<s>[INST] Evaluating $\iiint\frac{xyz\,dx\,dy\,dz}{\sqrt{a^2x^2+b^2y^2+c^2z^2}}$ over $\{(x,\,y,\,z)\in[0,\,\infty)^3\mid x^2+y^2+z^2\le R^2\}$ 
Evaluate $$\iiint\frac{xyz\,dx\,dy\,dz}{\sqrt{a^2x^2+b^2y^2+c^2z^2}}$$ for $a>b>c>0$ where $\Omega:=\{(x,\,y,\,z)\in[0,\,\infty)^3\mid x^2+y^2+z^2\le R^2\}$

What do we do with the denominator here?
 [/INST] Integrate in spherical coordinates
\begin{align}
&\iiint\frac{xyz}{\sqrt{a^2x^2+b^2y^2+c^2z^2}}\,dx\,dy\,dz\\
=& \int_0^{\pi/2} \int_0^{\pi/2}\int_0^R
\frac{r^4\sin \theta \cos\theta \sin^3\phi \cos\phi}
{\sqrt{ a^2\cos^2\theta\sin^2\phi+b^2\sin^2\theta\sin^2\phi +c^2\cos^2\phi}}dr d\theta d\phi\\
=& \int_0^{\pi/2} \int_0^{\pi/2}
\frac{R^5\sin^3\phi \cos\phi \ \sin \theta }
{5\sqrt{ a^2\sin^2\phi+ c^2\cos^2\phi +(b^2-a^2)\sin^2\phi\sin^2\theta}} d(\sin\theta)\ d\phi\\
 =& \int_0^{\pi/2} \frac{R^5\left(\sqrt{c^2-(c^2-{b^2})\sin^2\phi}-\sqrt{c^2-(c^2-{a^2})\sin^2\phi}\right)\sin\phi}{5(b^2-a^2)} d(\sin\phi)\\
=&\ \frac{R^5(ab+bc+ca)}{15(a+b)(b+c)(c+a)}
\end{align}
</s>