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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
