TITLE: Evaluation of $\int \frac{x\sin( \sqrt{ax^2+bx+c})}{ax^2+bx+c} \ dx\ $
QUESTION [5 upvotes]: How do we find $$\int  \frac{x\sin( \sqrt{ax^2+bx+c})}{ax^2+bx+c} \ dx\  $$  
NB: It is not mandatory that $ax^2+bx+c$ has    only a single root

REPLY [2 votes]: Hint:
$\int\dfrac{x\sin\sqrt{ax^2+bx+c}}{ax^2+bx+c}dx$
$=\int\dfrac{x}{ax^2+bx+c}\sum\limits_{n=0}^\infty\dfrac{(-1)^n(ax^2+bx+c)^{n+\frac{1}{2}}}{(2n+1)!}dx$
$=\int\sum\limits_{n=0}^\infty\dfrac{(-1)^nx(ax^2+bx+c)^{n-\frac{1}{2}}}{(2n+1)!}dx$