Given a vector of exponents $k=(k_1,\dots,k_n)$, I would like to get the exact value of the $n$-dimensional integral $$ I(k) = \int_{\mathbb{R}^n} \exp(-r) \prod_{i=1}^n x_i^{k_i} \;\text{d}x $$ where $r = \|x\|_2$, i.e., the monomials with exponents $k_i$ integrated over the entire space with the weight function $\exp(-r)$.
(Note that with the weight function $\exp(-r^2)$, the integral evaluates to $\prod_{i=1}^n \Gamma\left(\frac{k_i+1}{2}\right)$ if all $k_i$ are even, and 0 otherwise.)
Got it: $$ I(k) = \begin{cases} \left.2 \left(n + s - 1\right)! \prod_{i=1}^n\Gamma\left(\frac{k_i+1}{2}\right) \middle/ \Gamma\left(\frac{n + s}{2}\right)\right. \qquad&\text{ if all $k_i$ are even}\\ 0 &\text{ otherwise}. \end{cases} $$ with $s = \sum_{i=1}^n k_i$.