Simplify integral to beta or gamma function

Question

Can someone help me to simplify this integral by using beta or gamma functions:

$$ I= \int_{0}^\infty \left(1- \frac{a}{(b+c\cdot r^{-\alpha})^n}\right) r\,dr $$

Many thanks in advance.

Answer 1

Handle the term $r$ separately. Then, WLOG $b=1$ and with $s:=cr^{-\alpha}$, so that $dr\propto s^{-1-1/\alpha}ds$, we have

$$\int a(1+cr^{-\alpha})^{-n}\,r\,dr\propto\int(1+s)^{-n}\,s^{-1-2/\alpha}ds.$$