Consider the following Wishart distribution
$$ f({\bf W}) = \frac{ |{\bf W}|^{(n-p-1)/2} \exp\big[-\frac{1}{2}\text{tr}({\bf V}^{-1}{\bf W} ) \big] }{2^{np/2} |{\bf V}| \Gamma_p(\frac{n}{2})} \tag{1} $$
where ${\bf W}$ and ${\bf V}$ are $p \times p$ symmetric positive definite matrices, $|{\bf A}| \equiv det {\bf A}$, and $\Gamma_p$ is the multivariate gamma function . Calling $q \equiv p-1$, Let's write ${\bf V}$ as
$$ {\bf V} =\begin{pmatrix}a & b_1 & b_2 & ... & b_{q}\\\ b_1 & c_{11} & c_{12} & ... & c_{1q} \\\ b_2 & c_{21} & c_{22} & ... & c_{2 q}\\\ \vdots & \vdots & \vdots & \ddots & \vdots\\\ b_{q} & c_{q1} & c_{q2} & ... & c_{qq}\end{pmatrix}$$
Now, I would like to marginalize ($1$) over all the $c_{ij}$ with $i,j = 1,2,...q$. I have done the integration in the case where $p=2$ and ${\bf V}$ has the simple form
$$\begin{pmatrix}a & b\\\ b & c\end{pmatrix}$$
But now I would like to generalize that computation. I am pretty sure this must been done already in some book or paper but I haven't been able to find anything. Is there a general formula for arbitrary $p$ about this or a reference I can read?
With $p>2$ things gets soon very cumbersome, so tips on how to do it on my own in the case $p=3$ without it being a mess are welcome
