The question is related to my previous question about integrating the multinomial over the hypercube and the motivation for this question is the same, but the integral is a bit different. Here it is,

$$\int_{b}^{a}\cdots\int_{b}^{a} \left( \sum_{i=1}^{n}x_i^2\right)^k \left( \sum_{i=1}^{n}x_i\right)^m dx_1d x_2\dots dx_n.$$

Richard Stanley defltly solved the last integral, but this one does not appear to accept the same approach (the $x_i^2$ gets in the way).

I think I might be able to solve it using the partition function, but the solution is going to be terribly ugly. I am hoping that someone knows a tidy solution, or knows good references for solving integrals of this type.