Is there any hope to obtain an analytical expression for the following expectation value?
$$ \mathbb{E}\left\{ \frac{X^p}{\sum_{s\in \mathcal{S}} a_s X^s } \right\} $$
$p \in \mathbb{N}_+$ and $\mathcal{S} \subset \mathbb{N}_+$ is a set possibly containing $p$. $X$ is a random variable. Ideally Gaussian distribution (or even better Rayleigh) but if it makes calculations easier a different distribution may be possible.