Mathematica gives me the following solution to $\int{x^{-a} (b -cx^{-d})^e }$:


$$\int{x^{-a} (b -cx^{-d})^e dx} = -\frac{b^{e}x^{1-a} \, \_2F\_1\left(\frac{a-1}{d},-e;\frac{a+d-1}{d};\frac{c x^{-d}}{b}\right)}{a-1},$$
where $a,b,c,d,e > 0$.


I'd like to figure out what are the steps involved in this solution.


I started by finding the Taylor's expansion of the integrand, but it doesn't seem to be the right way to follow.



What are the steps involved in the solution to $\int{x^{-a} (b -cx^{-d})^e }dx$?
The proof starts by applying the change of variable $z=(c/b)x^{-d}$ and then using the following identity
$$\int (1-z)^{u} z^{v} dz = \frac{z^{v+1}}{v+1} \, \_2F\_1(-u, v+1; v+2; z), |x|<1,$$
the proof is concluded.


