How can I find the antiderivative of this complicated Bessel function?

[oh20elyf]

I am struggling to find the antiderivative of the following function:
$$f(x)=\frac{J_{0}(ax)J_{1}(bx) }{x+x^{4} } \\ J_{0},{~}J_{1} : Bessel{~}functions{~}of{~}the{~}first{~}kind\\ a, b: constants \\ F(x)=\int_{}^{} \! f(x) \, dx =?$$
Who can help?

 

[RUber]

I think you should use some principles from complex analysis.
$J_0(0) = 0$, $J_1(0) = 1$, So This function should be continuous around zero with a point discontinuity.
I would recommend using the residue formula for the roots of (1+ x^3) in the denominator.