I would like to evaluate the following integral analytically, but Mathematica does not give me an answer:
$$ \int_0^1 dr \ e^{(1-2r)x^2} \left[p(r,x) Y_0\left(2x^2\sqrt{r-r^2}\right)+q(r,x) \sqrt{r-r^2} \,Y_1\left(2x^2 \sqrt{r-r^2}\right)\right], $$
where
$$ \begin{align} p(r,x) &= -16x^4 r^4 + (32 x^4+8 x^2) r^3 - (26x^4 + 12 x^2) r^2 + (10x^4 + 4x^2) r - 2x^4, \\ q(r,x) &= -16x^4 r^3 + (24x^4 + 4x^2) r^2 - (16x^4 + 4x^2 - 8) r + 4x^4 + 3x^2 - 4, \end{align} $$
$x$ is positive, and $Y_n(z)$ indicates the Bessel function of the second kind.
Any ideas?