I need to evaluate the following matrix integral:
$\int_{P>0}\mathrm{det}(P)^{-1/2}\mathrm{det}(I+M\otimes P)^{-1/2}~_0F_1\left(\frac{d}{2}, -\frac{1}{4}SS^TP\right)~dP$
where $M$ and $P$ are $d\times d$ and positive-definite.
My first thought was to define $A=M\otimes P$ and make the change of variables $P\to A$. It's not too hard to show that
$dA = \sqrt{\mathrm{det}\left(J^TJ\right)}dP$
where
$J = (I_d\otimes K_{dd}\otimes I_d)(\mathrm{vec}(M)\otimes I_d)$
but now I'm unsure of how to proceed. I don't think that $A=M\otimes P$ has a convenient solution for $P$ in terms of $A$ and $M$, so just solving and substituting for $P$ in the rest of the integral doesn't seem to work.
Another approach could be to try and simplify $\mathrm{det}(I+M\otimes P)$, but I don't see any useful ways to do this. If we just had $\mathrm{det}(M\otimes P)$, then we could write it as $\mathrm{det}(M)^d\mathrm{det}(P)^d$ and not worry about any variable changes, but that's obviously not possible here.