FWIW :

```
sage: var("a")
a
sage: foo=1/((x-a)^(1/2)*x^(3/4))
sage: foo.integrate((x, 0, oo))
---------------------------------------------------------------------------
RuntimeError Traceback (most recent call last)
```

[ Snip... ]

```
ValueError: Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a>0)', see `assume?` for more details)
Is a positive, negative or zero?` arguments fails
```

Well...

```
sage: with assuming(real(a)<0): gee=foo.integrate((x, 0, oo)) ; gee
beta(1/4, 1/4)/(-a)^(1/4)
sage: gee.simplify_factorial()
gamma(1/4)^2/(sqrt(pi)*(-a)^(1/4))
```

Following Sage's suggestion gives us an usable result for `a<0`

(and fails for `a>=0`

, BTW).

Let's try someting else. Adding `algorithm="sympy"`

to `integrate arguments fails, because the result cannot (yet) be translated back to Sage. Try it manually :

```
sage: import sympy
sage: sympy.integrate(*map(sympy.sympify, (foo, (x, 0, oo))))
Piecewise((-a*exp(3*I*pi/4)*gamma(1/4)**2/(sqrt(pi)*polar_lift(a)**(5/4)), (Abs(arg(a) + pi) < pi) | (Ne(1/a, 0) & (Abs(arg(a) + pi) < pi) & Ne(Abs(arg(a) + pi), pi))), (Integral(1/(x**(3/4)*sqrt(-a + x)), (x, 0, oo)), True))
```

which ultimately means $ \frac{a e^{\frac{3 i \pi}{4}} \Gamma^{2}\left(\frac{1}{4}\right)}{\sqrt{\pi} \operatorname{polar_lift}^{\frac{5}{4}}{\left(a \right)}} $ if `abs(arg+pi)<pi`

, the latter reducing to `real(a)<0`

in I'm not confused.

Sympy fails to simplify the complicated expressions it uses (and cannot be translated to sage). But, FWIW :

```
sage: foo._mathematica_().Integrate((x, 0, oo))
ConditionalExpression[Gamma[1/4]^2/((-a)^(1/4)*Sqrt[Pi]), Re[a] < 0]
```

The real situation is therefore a tad moree complex than described.

Sage's can find an explicit form of the integral under a condition which it cannot find.

Sympy cannot find simplified expressions of its solution.

Mathematica agrees with Sage's integratin and seems to agree to Sympy's condition.

HTH,

Are you using an old version of sage ? On a recent one, the result is

which means that none of our integration engines can find a closed expression.

I am using the online version SageMath 9.1 at http://127.0.0.1:8888/notebooks... The value of the integral is a^(-1/4)*B(1/2,1/4) (with B the beta function, and a>0). The link with the beta function is straightforward by a change of variable.

sage 9.1 is very obsolete, but recent sage cannot do this integral

I may be wrong but it seems Sage 9.1 is the only version available on Jupyter.

Sage 10.3 (just released) runs without problem on Linux and on Windows>=10 nder WSL2. The recommended way to intall it is still compliing the source tarball (or, better, the

`git`

tree), but large progress have been made towards using Conda.