stuck with integration of $\frac{x^b}{c+x^b}$ where $b\in R_-$ and $c$ is constant

Question

I want to find out the integration of $$\int \frac{x^b}{c+x^b} $$ where $b\in R_-$(non-positive real number) and $c$ is some constant. I want closed-form expression for the above integration. It would be very helpful if one can help me out with this integration.

Answer 1

Using the trick of adding (+c-c) to the numerator we get: $$\int \frac{x^b\color{red}{+c-c}}{x^b+c}dx=\int dx-c\int\frac{dx}{x^b+c}=x-x\cdot{}_2F_1\left(1,\frac 1b;1+\frac 1b;-\frac{x^b}c\right)+constant$$ And this leads, according to Mathematica, to an ordinary Hypergeometric function.