Is there an analytic formula for fractional moments of multivariate normal distribution? $E(\prod_{i=1}^k x_i^{\nu_i})={?}$ where $X=(x_1,\ldots,x_k) \sim N_k(\mu, \Sigma)$, $\nu_i\in \mathcal{R}$ and $\nu_i>0$. I know there is one for univariate normal distributions, but can't find one for the multivariate case.

The formula for the univariate case I know of is $E[|x-\mu|^{\nu}]=\sigma^{\nu}\frac{2^{\nu/2}\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi}}$. I understand this is about the central absulate moment, but I am more interested in $E(\prod_{i=1}^k x_i^{\nu_i})$ or $E(\prod_{i=1}^k |x_i|^{\nu_i})$.