$$\int_{0}^{2\pi} \int_{0}^{1}r^5\sin^22\theta\left(1-r^2 \right)^2\sqrt{1+\left(1+ \cos^2\theta \right)36r^2 }\hspace{1mm}drd\theta$$
I tried integrating myself, spent many hours but could not figure out really anything, even used wolfram, it says time up!
Ignoring the $\theta$ and only considering $r$ basically this has the form $$r\cdot p(r^2)\cdot \sqrt{1+c\cdot r^2}$$ where $p$ is a polynomial. So you can try to substitute $t=1+c r^2$, so you end up with an integral over $\frac{1}{2c} p(t) \cdot \sqrt{t}$, which should pose no problem.
This still leaves the integral over $\theta$, which maybe you can attack with the usual bag of tricks, like using symmetries and so on.