Integral of product of irrational powers

[niterida]

Hello,

I'm trying to integrate a product of irrational "polynomials" that have arisen through dimensional regularization. As a warmup, I'm trying to understand how to integrate:

(w-a)^b (w-c)^d,

where b and d are arbitrary numbers, possibly irrational.

Mathematica can solve this indefinite integral, getting:

(((a - w)/(a - c))^-b (-a + w)^b (-c + w)^(1 + d)
Hypergeometric2F1[-b, 1 + d, 2 + d, (-c + w)/(a - c)])/(1 + d)

Can anyone explain how the answer is obtained?

Also, if there are any good tables (in print or online) or references on the subject of integrating
irrational powers, I would be very interested. Thank you.

 

[fresh_42]

Here are quite a few formulas for integration:
https://de.wikibooks.org/wiki/Formelsammlung_Mathematik:_IntegraleIt's not English but the formulas speak for themselves.