# Marginal Distribution of Matrix Normal with Two Inverse Wisharts

Say I have a Matrix-Normal distribution and two Inverse Wishart Distributions

$$X \sim MN_{p\times n}(0, \Sigma, \Omega)$$ $$\Sigma \sim IW(a, A)$$ $$\Omega \sim IW(b, B)$$ where $$a$$ and $$b$$ are degrees of freedom and $$A$$ and $$B$$ are scale matrices.

I know that the marginal distribution of $$p(X, \Omega)$$ is matrix-T and given by $$T_{p\times n} (a, 0, A, \Omega)$$. What is the marginal distribution of $$p(X)$$?

In other words, what is the marginal distribution when you marginalize over both inverse Wishart random variables? Is there a name for this distribution?