Integration with quadratic square root

Question

What could be the techniques we need to use to solve this integration

$\int\dfrac{s^2\sin^2\left(s\sqrt{ as^2+bs+c}\right)}{as^2+bs+c}ds$ ?

Main issue here is the term inside $\sin^2()$. Very difficult to make it in proper form

Answer 1

Hint:

$\int\dfrac{s^2\sin^2\left(s\sqrt{as^2+bs+c}\right)}{as^2+bs+c}ds$

$=\int\dfrac{s^2}{as^2+bs+c}\dfrac{1-\cos\left(2s\sqrt{as^2+bs+c}\right)}{2}ds$

$=-\int\dfrac{s^2}{2(as^2+bs+c)}\sum\limits_{n=1}^\infty\dfrac{(-1)^n4^ns^{2n}(as^2+bs+c)^n}{(2n)!}ds$

$=\int\sum\limits_{n=1}^\infty\dfrac{(-1)^{n+1}2^{2n-1}s^{2n+2}(as^2+bs+c)^{n-1}}{(2n)!}ds$