Triple Integrals & Coordinate Systems · Superchallenge problem

A solid unit cube $[0,1]^3$ has constant density $1$. Find its moment of inertia about the line through $(0,0,0)$ and $(1,1,1)$ (a space diagonal), that is, $\displaystyle\iiint d^2\,dV$ where $d$ is the distance from a point to that line.