Integral over positive definite matrix, $\int_{A > 0} (I + A)^{-2} |A|^{(n-p-1)/2} |I + A|^{-(n+\nu)/2} dA$.

Question

Let $\mathbf{A}$ be a $p \times p$ positive definite matrix.

$$ \int_{\mathbf{A} > 0} (\mathbf{I} + \mathbf{A})^{-2} |\mathbf{A}|^{(n-p-1)/2} |\mathbf{I} + \mathbf{A}|^{-(n+\nu)/2} d\mathbf{A} = \quad ? $$

This question is related to https://stats.stackexchange.com/q/525851/88700.

See also https://dlmf.nist.gov/35.3.8.