Integration of Gamma PDF fails

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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

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.