Integration of Gamma PDF fails

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WoksForMe(TM) on 9.3.rc5 ... provided you provide the right information :

sage: var("x, alpha, beta", domain="positive")                                  
(x, alpha, beta)
sage: assume(alpha,"noninteger")                                                
sage: integrate(x^(alpha-1)*e^(-beta*x),x,0,oo)                                 
gamma(alpha)/beta^alpha

From sage.symbolic.assumptions? :

Here is the list of acceptable features::

    sage: maxima('features')
    [integer,noninteger,even,odd,rational,irrational,real,imaginary,complex,analytic,increasing,decreasing,oddfun,evenfun,posfun,constant,commutative,lassociative,rassociative,symmetric,antisymmetric,integervalued]

As far as I recall, one of the examples of assuming uses the integration of the gamma distribution to illustrate how to (temporarily) assume alpha being noninteger.

This might only partially answer your question, but hopefully it helps.

Declaring a and b to be positive integers:

sage: a, b = SR.var('a, b', domain='positive')
sage: assume("a, b, 'integer'")

sage: assumptions()
[b > 0, a > 0, a is integer, b is integer]

and declaring f as a function of a single variable

sage: f(x) = x^(a - 1) * e^(-b*x)

we get:

sage: f.integrate(x, 0, oo)
gamma(a)/b^a