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answered 2021-05-17 19:09:07 +0100

Emmanuel Charpentier gravatar image

The problem is not the warnings, but our conversion to Sage : 

sage: Ex=(-b*x+2)^(5/2)*x^(1/2)
sage: %time GI=libgiac(Ex).integrate(libgiac(x))
CPU times: user 17.1 s, sys: 412 ms, total: 17.6 s
Wall time: 17.6 s
sage: GI
GI
1/b*(2*b^3*abs(b)/b^2*(2*(((-90*b^11*1/1440/b^14*sqrt(-b*x+2)*sqrt(-b*x+2)+750*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*x+2)-2445*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*x+2)+4185*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-35/8/b^2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-12*b^2*abs(b)/b^2*(2*((12*b^5*1/144/b^7*sqrt(-b*x+2)*sqrt(-b*x+2)-78*b^5*1/144/b^7)*sqrt(-b*x+2)*sqrt(-b*x+2)+198*b^5*1/144/b^7)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-5/2/b/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-24*b/b*abs(b)/b^2*(2*(1/8*sqrt(-b*x+2)*sqrt(-b*x+2)-5/8)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)+6*b/4/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-16*abs(b)/b^2*(1/2*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-2*b/2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2)))))

After a bit of exploration ('not shown), it turns out that giacs's ln is problematic. So we can work around this. Awful version :

sage: SR(str(GI).replace("ln","log")).simplify_full()
1/24*((6*b^3*x^3 - 34*b^2*x^2 + 59*b*x - 15)*sqrt(b^2*x)*sqrt(-b*x + 2) - 30*sqrt(-b)*log(abs(sqrt(-b*x + 2)*sqrt(-b) - sqrt(b^2*x))))/(b*abs(b))

$$ \frac{{\left(6 \, b^{3} x^{3} - 34 \, b^{2} x^{2} + 59 \, b x - 15\right)} \sqrt{b^{2} x} \sqrt{-b x + 2} - 30 \, \sqrt{-b} \log\left({\left| \sqrt{-b x + 2} \sqrt{-b} - \sqrt{b^{2} x} \right|}\right)}{24 \, b {\left| b \right|}} $$

In fact, all the Sage-accessible integrators need a bit nudging to obtain usable results. Meditate the results of the following code (unLaTeXable by this site's tools) :

var("b")
Ex = (-b*x+2)^(5/2)*x^(1/2)
t0 = systime()
with assuming(b<0): MN = Ex.integrate(x).simplify_full()
M0 = Ex.subs(b==0).integrate(x).simplify_full()
with assuming(b>0): MP = Ex.integrate(x).simplify_full()
M = cases([(b<0, MN), (b==0, M0), (b>0, MP)]).function(x)
tM=systime()
S = Ex.integrate(x, algorithm="sympy").function(x)
tS= systime()
G = SR(str(libgiac(Ex).integrate(libgiac(x))).replace("ln","log"))\
    .simplify_full().function(x)
tG = systime()
Fl = [u.function(x) for u in Ex.integrate(x, algorithm="fricas")]
tF = systime()
Sqrt = function("Sqrt")
MM = mathematica.Integrate(Ex, x)\
                .sage().substitute_function(Sqrt,sqrt).function(x)
tMM = systime()

TR = table(rows=[["Sage", round(tM-t0,3), M],
                 ["Sympy", round(tS-tM, 3), S],
                 ["Giac", round(tG-tS, 3), G],
                 ["Fricas", round(tF-tG, 3), matrix([[Fl[0]],[Fl[1]]])],
                 ["Mathematica", round(tMM-tF, 3), MM]],
           header_row=["Integrator","time", "result"])

Comments later (France's curfew ahead..).

The problem is not the warnings, but our conversion to Sage : 

sage: Ex=(-b*x+2)^(5/2)*x^(1/2)
sage: %time GI=libgiac(Ex).integrate(libgiac(x))
CPU times: user 17.1 s, sys: 412 ms, total: 17.6 s
Wall time: 17.6 s
sage: GI
GI
1/b*(2*b^3*abs(b)/b^2*(2*(((-90*b^11*1/1440/b^14*sqrt(-b*x+2)*sqrt(-b*x+2)+750*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*x+2)-2445*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*x+2)+4185*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-35/8/b^2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-12*b^2*abs(b)/b^2*(2*((12*b^5*1/144/b^7*sqrt(-b*x+2)*sqrt(-b*x+2)-78*b^5*1/144/b^7)*sqrt(-b*x+2)*sqrt(-b*x+2)+198*b^5*1/144/b^7)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-5/2/b/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-24*b/b*abs(b)/b^2*(2*(1/8*sqrt(-b*x+2)*sqrt(-b*x+2)-5/8)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)+6*b/4/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-16*abs(b)/b^2*(1/2*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-2*b/2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2)))))

After a bit of exploration ('not shown), it turns out that giacs's ln is problematic. So we can work around this. Awful version :

sage: SR(str(GI).replace("ln","log")).simplify_full()
1/24*((6*b^3*x^3 - 34*b^2*x^2 + 59*b*x - 15)*sqrt(b^2*x)*sqrt(-b*x + 2) - 30*sqrt(-b)*log(abs(sqrt(-b*x + 2)*sqrt(-b) - sqrt(b^2*x))))/(b*abs(b))

$$ \frac{{\left(6 \, b^{3} x^{3} - 34 \, b^{2} x^{2} + 59 \, b x - 15\right)} \sqrt{b^{2} x} \sqrt{-b x + 2} - 30 \, \sqrt{-b} \log\left({\left| \sqrt{-b x + 2} \sqrt{-b} - \sqrt{b^{2} x} \right|}\right)}{24 \, b {\left| b \right|}} $$

In fact, all the Sage-accessible integrators need a bit nudging to obtain usable results. Meditate the results of the following code (unLaTeXable by this site's tools) :

var("b")
Ex = (-b*x+2)^(5/2)*x^(1/2)
t0 = systime()
with assuming(b<0): MN = Ex.integrate(x).simplify_full()
M0 = Ex.subs(b==0).integrate(x).simplify_full()
with assuming(b>0): MP = Ex.integrate(x).simplify_full()
M = cases([(b<0, MN), (b==0, M0), (b>0, MP)]).function(x)
tM=systime()
S = Ex.integrate(x, algorithm="sympy").function(x)
tS= systime()
G = SR(str(libgiac(Ex).integrate(libgiac(x))).replace("ln","log"))\
    .simplify_full().function(x)
tG = systime()
Fl = [u.function(x) for u in Ex.integrate(x, algorithm="fricas")]
tF = systime()
Sqrt = function("Sqrt")
MM = mathematica.Integrate(Ex, x)\
                .sage().substitute_function(Sqrt,sqrt).function(x)
tMM = systime()

TR = table(rows=[["Sage", round(tM-t0,3), M],
                 ["Sympy", round(tS-tM, 3), S],
                 ["Giac", round(tG-tS, 3), G],
                 ["Fricas", round(tF-tG, 3), matrix([[Fl[0]],[Fl[1]]])],
                 ["Mathematica", round(tMM-tF, 3), MM]],
           header_row=["Integrator","time", "result"])

Comments later (France's curfew ahead..).

EDIT :

As announced, a few comments :

  • Fricas' answer is (probably) incomplete : both expressions rederive to Ex, but their validity limits are not stated. This is to be explored within the original Fricas interface, with a probable followup ticket.

  • Giac library's answer as well as Sympy's in the case $\frac{\left|bx\right|}{2}>1$ are a figment of their respective authors' imagination: the derivatives of the proposed expression cannot be proved equal to Ex (but I have not computed their quotients or ratios, just plotted a few numerical cases). Deriving the other expressions with respect to x give expressions which can be proved equal to Ex.

  • It is probably not a coincidence if these two incorrect answers are obtained after a much longer time than "correct" ones...

  • Using cases expressions proved surprisingly heavy. For example, the subs and plot methods don't work at all. The diff method does surprising things. This has to be enhanced : ticket ahoy ! (But this one is difficult to report correctly...).

  • A peek at the answer given by Wolframl alpha gives hints on the origin of the different antiderivative forms given bu=y the different CASes.

In conclusion :

CASes are a big help, but you MUST check their answers by any means available. And a great strength of Sage is the availability of a common framework and interface to various CASes.

The problem is not the warnings, but our conversion to Sage : 

sage: Ex=(-b*x+2)^(5/2)*x^(1/2)
sage: %time GI=libgiac(Ex).integrate(libgiac(x))
CPU times: user 17.1 s, sys: 412 ms, total: 17.6 s
Wall time: 17.6 s
sage: GI
GI
1/b*(2*b^3*abs(b)/b^2*(2*(((-90*b^11*1/1440/b^14*sqrt(-b*x+2)*sqrt(-b*x+2)+750*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*x+2)-2445*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*x+2)+4185*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-35/8/b^2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-12*b^2*abs(b)/b^2*(2*((12*b^5*1/144/b^7*sqrt(-b*x+2)*sqrt(-b*x+2)-78*b^5*1/144/b^7)*sqrt(-b*x+2)*sqrt(-b*x+2)+198*b^5*1/144/b^7)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-5/2/b/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-24*b/b*abs(b)/b^2*(2*(1/8*sqrt(-b*x+2)*sqrt(-b*x+2)-5/8)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)+6*b/4/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-16*abs(b)/b^2*(1/2*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-2*b/2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2)))))

After a bit of exploration ('not shown), it turns out that giacs's ln is problematic. So we can work around this. Awful version :

sage: SR(str(GI).replace("ln","log")).simplify_full()
1/24*((6*b^3*x^3 - 34*b^2*x^2 + 59*b*x - 15)*sqrt(b^2*x)*sqrt(-b*x + 2) - 30*sqrt(-b)*log(abs(sqrt(-b*x + 2)*sqrt(-b) - sqrt(b^2*x))))/(b*abs(b))

$$ \frac{{\left(6 \, b^{3} x^{3} - 34 \, b^{2} x^{2} + 59 \, b x - 15\right)} \sqrt{b^{2} x} \sqrt{-b x + 2} - 30 \, \sqrt{-b} \log\left({\left| \sqrt{-b x + 2} \sqrt{-b} - \sqrt{b^{2} x} \right|}\right)}{24 \, b {\left| b \right|}} $$

In fact, all the Sage-accessible integrators need a bit nudging to obtain usable results. Meditate the results of the following code (unLaTeXable by this site's tools) :

var("b")
Ex = (-b*x+2)^(5/2)*x^(1/2)
t0 = systime()
with assuming(b<0): MN = Ex.integrate(x).simplify_full()
M0 = Ex.subs(b==0).integrate(x).simplify_full()
with assuming(b>0): MP = Ex.integrate(x).simplify_full()
M = cases([(b<0, MN), (b==0, M0), (b>0, MP)]).function(x)
tM=systime()
S = Ex.integrate(x, algorithm="sympy").function(x)
tS= systime()
G = SR(str(libgiac(Ex).integrate(libgiac(x))).replace("ln","log"))\
    .simplify_full().function(x)
tG = systime()
Fl = [u.function(x) for u in Ex.integrate(x, algorithm="fricas")]
tF = systime()
Sqrt = function("Sqrt")
MM = mathematica.Integrate(Ex, x)\
                .sage().substitute_function(Sqrt,sqrt).function(x)
tMM = systime()

TR = table(rows=[["Sage", round(tM-t0,3), M],
                 ["Sympy", round(tS-tM, 3), S],
                 ["Giac", round(tG-tS, 3), G],
                 ["Fricas", round(tF-tG, 3), matrix([[Fl[0]],[Fl[1]]])],
                 ["Mathematica", round(tMM-tF, 3), MM]],
           header_row=["Integrator","time", "result"])

Comments later (France's curfew ahead..).

EDIT :

As announced, a few comments :

  • Fricas' answer is (probably) incomplete : both expressions rederive to Ex, but their validity limits are not stated. This is to be explored within the original Fricas interface, with a probable followup ticket.

  • Giac library's answer as well as Sympy's in the case $\frac{\left|bx\right|}{2}>1$ are a figment of their respective authors' imagination: the derivatives of the proposed expression cannot be proved equal to Ex (but I have not computed their quotients or ratios, just plotted a few numerical cases). Deriving the other expressions with respect to x give expressions which can be proved equal to Ex.

  • It is probably not a coincidence if these two incorrect answers are obtained after a much longer time than "correct" ones...

  • Using cases expressions proved surprisingly heavy. For example, the subs and plot methods don't work at all. The diff method does surprising things. This has to be enhanced : ticket ahoy ! (But this one is difficult to report correctly...).

  • A peek at the answer given by Wolframl alpha gives hints on the origin of the different antiderivative forms given bu=y the different CASes.

In conclusion :

CASes are a big help, but you MUST check their answers by any means available.available (including numerical ones, which cannot prove you right but can prove you wrong) And a great strength of Sage is the availability of a common framework and interface to various CASes.

The problem is not the warnings, but our conversion to Sage : 

sage: Ex=(-b*x+2)^(5/2)*x^(1/2)
sage: %time GI=libgiac(Ex).integrate(libgiac(x))
CPU times: user 17.1 s, sys: 412 ms, total: 17.6 s
Wall time: 17.6 s
sage: GI
GI
1/b*(2*b^3*abs(b)/b^2*(2*(((-90*b^11*1/1440/b^14*sqrt(-b*x+2)*sqrt(-b*x+2)+750*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*x+2)-2445*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*x+2)+4185*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-35/8/b^2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-12*b^2*abs(b)/b^2*(2*((12*b^5*1/144/b^7*sqrt(-b*x+2)*sqrt(-b*x+2)-78*b^5*1/144/b^7)*sqrt(-b*x+2)*sqrt(-b*x+2)+198*b^5*1/144/b^7)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-5/2/b/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-24*b/b*abs(b)/b^2*(2*(1/8*sqrt(-b*x+2)*sqrt(-b*x+2)-5/8)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)+6*b/4/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-16*abs(b)/b^2*(1/2*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-2*b/2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2)))))

After a bit of exploration ('not shown), it turns out that giacs's ln is problematic. So we can work around this. Awful version :

sage: SR(str(GI).replace("ln","log")).simplify_full()
1/24*((6*b^3*x^3 - 34*b^2*x^2 + 59*b*x - 15)*sqrt(b^2*x)*sqrt(-b*x + 2) - 30*sqrt(-b)*log(abs(sqrt(-b*x + 2)*sqrt(-b) - sqrt(b^2*x))))/(b*abs(b))

$$ \frac{{\left(6 \, b^{3} x^{3} - 34 \, b^{2} x^{2} + 59 \, b x - 15\right)} \sqrt{b^{2} x} \sqrt{-b x + 2} - 30 \, \sqrt{-b} \log\left({\left| \sqrt{-b x + 2} \sqrt{-b} - \sqrt{b^{2} x} \right|}\right)}{24 \, b {\left| b \right|}} $$

In fact, all the Sage-accessible integrators need a bit nudging to obtain usable results. Meditate the results of the following code (unLaTeXable by this site's tools) :

var("b")
Ex = (-b*x+2)^(5/2)*x^(1/2)
t0 = systime()
with assuming(b<0): MN = Ex.integrate(x).simplify_full()
M0 = Ex.subs(b==0).integrate(x).simplify_full()
with assuming(b>0): MP = Ex.integrate(x).simplify_full()
M = cases([(b<0, MN), (b==0, M0), (b>0, MP)]).function(x)
tM=systime()
S = Ex.integrate(x, algorithm="sympy").function(x)
tS= systime()
G = SR(str(libgiac(Ex).integrate(libgiac(x))).replace("ln","log"))\
    .simplify_full().function(x)
tG = systime()
Fl = [u.function(x) for u in Ex.integrate(x, algorithm="fricas")]
tF = systime()
Sqrt = function("Sqrt")
MM = mathematica.Integrate(Ex, x)\
                .sage().substitute_function(Sqrt,sqrt).function(x)
tMM = systime()

TR = table(rows=[["Sage", round(tM-t0,3), M],
                 ["Sympy", round(tS-tM, 3), S],
                 ["Giac", round(tG-tS, 3), G],
                 ["Fricas", round(tF-tG, 3), matrix([[Fl[0]],[Fl[1]]])],
                 ["Mathematica", round(tMM-tF, 3), MM]],
           header_row=["Integrator","time", "result"])

Comments later (France's curfew ahead..).

EDIT :

As announced, a few comments :

  • Fricas' answer is (probably) incomplete : both expressions rederive to Ex, but their validity limits are not stated. This is to be explored within the original Fricas interface, with a probable followup ticket.

  • Giac library's answer as well as Sympy's in the case $\frac{\left|bx\right|}{2}>1$ are a figment of their respective authors' imagination: the derivatives of the proposed expression cannot be proved equal to Ex (but I have not computed their quotients or ratios, just plotted a few numerical cases). Deriving the other expressions with respect to x give expressions which can be proved equal to Ex.

  • It is probably not a coincidence if these two incorrect answers are obtained after a much longer time than "correct" ones...

  • Using cases expressions proved surprisingly heavy. For example, the subs and plot methods don't work at all. The diff method does surprising things. This has to be enhanced : ticket ahoy ! (But this one is difficult to report correctly...).

  • A peek at the answer given by Wolframl alpha gives hints on the origin of the different antiderivative forms given bu=y the different CASes.

  • <rant> Mathjax is no sufficient substitute to $\LaTeX$...</rant>.

In conclusion :

CASes are a big help, but you MUST check their answers by any means available (including numerical ones, which cannot prove you right but can prove you wrong) And a great strength of Sage is the availability of a common framework and interface to various CASes.

The problem is not the warnings, but our conversion to Sage : 

sage: Ex=(-b*x+2)^(5/2)*x^(1/2)
sage: %time GI=libgiac(Ex).integrate(libgiac(x))
CPU times: user 17.1 s, sys: 412 ms, total: 17.6 s
Wall time: 17.6 s
sage: GI
GI
1/b*(2*b^3*abs(b)/b^2*(2*(((-90*b^11*1/1440/b^14*sqrt(-b*x+2)*sqrt(-b*x+2)+750*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*x+2)-2445*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*x+2)+4185*b^11*1/1440/b^14)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-35/8/b^2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-12*b^2*abs(b)/b^2*(2*((12*b^5*1/144/b^7*sqrt(-b*x+2)*sqrt(-b*x+2)-78*b^5*1/144/b^7)*sqrt(-b*x+2)*sqrt(-b*x+2)+198*b^5*1/144/b^7)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-5/2/b/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-24*b/b*abs(b)/b^2*(2*(1/8*sqrt(-b*x+2)*sqrt(-b*x+2)-5/8)*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)+6*b/4/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2))))-16*abs(b)/b^2*(1/2*sqrt(-b*x+2)*sqrt(-b*(-b*x+2)+2*b)-2*b/2/sqrt(-b)*ln(abs(sqrt(-b*(-b*x+2)+2*b)-sqrt(-b)*sqrt(-b*x+2)))))

After a bit of exploration ('not shown), it turns out that giacs's ln is problematic. So we can work around this. Awful version :

sage: SR(str(GI).replace("ln","log")).simplify_full()
1/24*((6*b^3*x^3 - 34*b^2*x^2 + 59*b*x - 15)*sqrt(b^2*x)*sqrt(-b*x + 2) - 30*sqrt(-b)*log(abs(sqrt(-b*x + 2)*sqrt(-b) - sqrt(b^2*x))))/(b*abs(b))

$$ \frac{{\left(6 \, b^{3} x^{3} - 34 \, b^{2} x^{2} + 59 \, b x - 15\right)} \sqrt{b^{2} x} \sqrt{-b x + 2} - 30 \, \sqrt{-b} \log\left({\left| \sqrt{-b x + 2} \sqrt{-b} - \sqrt{b^{2} x} \right|}\right)}{24 \, b {\left| b \right|}} $$

In fact, all the Sage-accessible integrators need a bit nudging to obtain usable results. Meditate the results of the following code (unLaTeXable by this site's tools) :

var("b")
Ex = (-b*x+2)^(5/2)*x^(1/2)
from time import time as systime
t0 = systime()
with assuming(b<0): MN = Ex.integrate(x).simplify_full()
M0 = Ex.subs(b==0).integrate(x).simplify_full()
with assuming(b>0): MP = Ex.integrate(x).simplify_full()
M = cases([(b<0, MN), (b==0, M0), (b>0, MP)]).function(x)
tM=systime()
S = Ex.integrate(x, algorithm="sympy").function(x)
tS= systime()
G = SR(str(libgiac(Ex).integrate(libgiac(x))).replace("ln","log"))\
    .simplify_full().function(x)
tG = systime()
Fl = [u.function(x) for u in Ex.integrate(x, algorithm="fricas")]
tF = systime()
Sqrt = function("Sqrt")
MM = mathematica.Integrate(Ex, x)\
                .sage().substitute_function(Sqrt,sqrt).function(x)
tMM = systime()

TR = table(rows=[["Sage", round(tM-t0,3), M],
                 ["Sympy", round(tS-tM, 3), S],
                 ["Giac", round(tG-tS, 3), G],
                 ["Fricas", round(tF-tG, 3), matrix([[Fl[0]],[Fl[1]]])],
                 ["Mathematica", round(tMM-tF, 3), MM]],
           header_row=["Integrator","time", "result"])

Comments later (France's curfew ahead..).

EDIT :

As announced, a few comments :

  • Fricas' answer is (probably) incomplete : both expressions rederive to Ex, but their validity limits are not stated. This is to be explored within the original Fricas interface, with a probable followup ticket.

  • Giac library's answer as well as Sympy's in the case $\frac{\left|bx\right|}{2}>1$ are a figment of their respective authors' imagination: the derivatives of the proposed expression cannot be proved equal to Ex (but I have not computed their quotients or ratios, just plotted a few numerical cases). Deriving the other expressions with respect to x give expressions which can be proved equal to Ex.

  • It is probably not a coincidence if these two incorrect answers are obtained after a much longer time than "correct" ones...

  • Using cases expressions proved surprisingly heavy. For example, the subs and plot methods don't work at all. The diff method does surprising things. This has to be enhanced : ticket ahoy ! (But this one is difficult to report correctly...).

  • A peek at the answer given by Wolframl alpha gives hints on the origin of the different antiderivative forms given bu=y the different CASes.

  • <rant> Mathjax is no sufficient substitute to $\LaTeX$...</rant>.

In conclusion :

CASes are a big help, but you MUST check their answers by any means available (including numerical ones, which cannot prove you right but can prove you wrong) And a great strength of Sage is the availability of a common framework and interface to various CASes.

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