General formula for $ \int_{0}^{1} \cdots \int_{0}^{1} \frac{x_1^q + \cdots + x_n^q}{x_1^p + \cdots + x_n^p} \, \mathrm{d}x_1 \cdots \mathrm{d}x_n $

Question

Recently, reading this problem, I found out this beautiful property

$ \lim_{n\to \infty} \int_{0}^{1} \cdots \int_{0}^{1} \frac{x_1^q + \cdots + x_n^q}{x_1^p + \cdots + x_n^p} \, \mathrm{d}x_1 \cdots \mathrm{d}x_n =\frac{p+1}{q+1} $

Is it known a general formula for that multiple integral when we fix $ n, p $ and $q$ as positive integers? Otherwise is it possible to have a general formula if $q=2, p=1$ and $n$ is a fixed positive integer?