Integral $ \int_{0}^{D} x \cdot (\sqrt{1+x^a})^{b}\,dx$

Question

do you have any ideas on solving the following integrals: $$ \int_{0}^{D} x \cdot (\sqrt{1+x^a})^{b}\,dx$$ where $b$ are positive integers and a is also positive.

For the special case of $a=2$, it is straightforward to solve; however for more general values, it is really tricky and change of RV using $\sqrt{1+x^a} = y$ seems not to make the integral easier.

Base on the requirements to solve binomial integral using elementary functions, there should be no solution to the integral using elementary functions; but how about using hypergeometric functions?

Thanks.

Answer 1

This is an example of a binomial integral. It can be rewritten as $$\int_{0}^{D}x^1\left(1+x^a\right)^{b/2}\,\mathrm{d}x$$ See the link for how to solve it