To me this integral is just begging to be written as a triple integral. But I can't think of a way of doing it that will make evaluating the integral any easier.
I'm pretty sure you meant .
Interestingly, Maple returns the wrong answers for different values of the parameters while Wolfram Alpha returns the correct values. Like for p = 3 and q =2, Maple returns a negative value of which is clearly absurd when you look at the graph.