Let $\vec r_1$, $\vec r_2$, and $\vec r$ be vectors in the Cartesian space.
The spherical average $\int |\vec r_1 -\vec r| \, d\Omega$ (where $\Omega$ refers to $\vec r$) readily evaluates to $\frac{2\pi}{3 r_1 r} \,[(r_1+r)^3-|r_1-r|^3]$.
Is there a closed-form expression for $\int |\vec r_1 -\vec r| |\vec r_2 -\vec r| \, d\Omega$ in the general case of $\vec r_1$ not being collinear with $\vec r_2$?