I'm looking for a way to symbolically integrate the following expression (the $k_i$ are integer constants):
Using a generalized form of the identity
I was able to express it without the cumbersome area condition of the original expression:
I've tried expanding the expression
using the multinomial theorem, in order to eventually pull the sum outside the integral and make an integration step, but it didn't look very promising, the formulas just kept getting larger and larger, with not obvious pattern emerging which I could use to iterate the process for all the integrals.
I realize that in the end, all that will ever come out of these formulas are multivariate polynomials, so in theory, it should be no problem to integrate this stuff symbolically. I just can't find a way to express the result concisely in standard mathematical notation.
Given that the original expression seems simple enough, I figured that there might be an easier way of approaching this problem, or that maybe it has already been solved.
So, does anyone know how to symbolically integrate the given expression?