Let $\mathbf{A}$ be symmetric positive definite matrices of dimension $(p \times p)$ and $a$, $b$ be positive reals.
I know that the following equality should hold $$ \int_{\mathbf{A} > 0} \mathbf{A} \, \text{det}(\mathbf{A})^\frac{a - p - 1}{2}(b + \text{trace}(\mathbf{A}))^{-\frac{b+ap}{2}} d\mathbf{A} = \frac{\Gamma_p(a/2) \Gamma(b/2) b^{b/2+1} a}{\Gamma((b+ap)/2) (b-2)} \mathbf{I}. $$
Has anyone seen this integral before and knows where I can find out more about it and related integrals?