I have come across an integral of the form $$\int_{b}^{a}\cdots\int_{b}^{a} \left( \sum_{i=1}^{n}x_i\right)^mdx_1d x_2\dots dx_n.$$ I have a solution that makes use of the partition function, but I feel there should be a much nicer solution and I'm sure this has been looked at before. Does anybody know a reference?

**Motivation:**
This integral has appeared whilst trying to compute moments of the Voronoi cell of the lattice $A_n$ (see page 462 of Sphere Packings, Lattices and Groups)