Multivariate Normal Product Distribution

Question

I am looking for multivariate case of a distribution of a product of two normally distributed variables X and Y. The variables are independent. Something similar to this:

http://mathworld.wolfram.com/NormalProductDistribution.html

Except:

1. Both X and Y are vectors and Gaussians are multivariate $N(\mu_x,\Sigma_x)$ and $N(\mu_y,\Sigma_y)$
2. Both distributions have non-zero respective means $\mu_x$ and $\mu_y$