Integration of a and b integral

Question

I'm trying to solve this integral $$\int \frac{x^a}{1+x^b}dx $$ I'm not sure how I should approach this.I've looked up for similar integrals but nothing seems to apply to this.Any help?I'm thinking of Leibniz rule but not sure how to apply

Answer 1

Welcome to the world of hypergeometric functions $$\int \frac{x^a}{1+x^b}dx=\frac{x^{a+1} }{a+1}\, _2F_1\left(1,\frac{a+1}{b};\frac{a+b+1}{b};-x^b\right)$$

Edit

It is possible that, without noticing it, you already used them. Being simplistic, use the well known series $$\frac 1{1+t}=\sum_{n=0}^\infty (-1)^n t^n$$ Make $t=x^b$ to get $$\frac{x^a}{1+x^b}=\sum_{n=0}^\infty (-1)^n x^{a+b n}$$ $$\int \frac{x^a}{1+x^b}dx=\sum_{n=0}^\infty \frac{(-1)^n x^{a+b n+1}}{a+b n+1}$$