Change of Variables & the Jacobian · Superchallenge problem

Let $D$ be the solid of points with $z > 0$, $1 \le z^2 - x^2 - y^2 \le 4$ and $\sqrt{x^2 + y^2} \le \dfrac{3z}{5}$. Evaluate $\displaystyle\iiint_D \dfrac{dV}{\left(z^2 - x^2 - y^2\right)^{3/2}}$.