Contour Integration of $\int \frac{d^3 x}{(a^2 + r^2)^3}$

Question

I'm working through the section on Contour Integration in Mathews and Walker's Mathematical Methods of Physics, and there are two integrals that I'm stumped on (I asked about one of them earlier). The other one is

$\int \frac{d^3 x}{(a^2 + r^2)^3}$

which is tripping me up because I'm not sure how to deal with the volume element. Would I have to draw a 3-dimensional contour? I have a feeling I'm overthinking this, any help would be appreciated.

Answer 1

Switch to spherical coordinates. Then the integral becomes $$\int d\Omega\int r^2dr\frac1{(a^2+r^2)^3}$$