It's well known how to perform integrals of the type (link)
$$I=\int n_in_jn_k\, d\Omega$$
where $n_i$ are the components of a 3D unit vector and $d\Omega$ is the solid angle. I have encountered integrals of the this type (the one I found actually has 8 on the numerator and 10 in the denoinator)
$$I=\int \frac{n_i n_j}{n_k n_l n_m n_o } d\Omega$$
and I wonder if there exist a general solution as in the first case. Clearly, we get a lot of cancelations when we have odd powers, but I feel it can get very messy without some general approach.