Given 2 ellpsoids, the outer with semi-major axes $A,B,C$ and the inner with semi-major axes $a,b,$ and $c$. I am supposed to integrate $\frac{1}{(\text{current volume})^3}$ (the volume at the current radius, not constant over the integral; (current volume) $=\frac{4\pi}{3}r^3$ for a sphere) over the volume between the ellipsoids. I tried mapping the ellipsoids to separate unit spheres, i.e.
$8 ABC \int_{0}^{\pi/2}\int_{0}^{\pi/2}\int_0^1\frac{1}{(\text{current volume})^3}r^2\sin\theta \,dr\,d\theta\,d\phi$
using octant symmetry and the Jacobian $|J|=ABC$ for the outer ellipsoid.
I became stuck here due to a lack a formulation for the current volume and also due to convergence problems of $\frac{1}{r}$ evaluated at zero. Is my approach sound? Is there a parameterization that allows the solution of this problem?