I am stuck in an integral problem:

$\int_{\mathbb{R}^d}(\det(\mathbf{A}+k(\mathbf{x}-\mathbf{y}_0)(\mathbf{x}-\mathbf{y}_0)^T))^{-r_1}\prod_{i=1}^n(\det(\mathbf{B}+(\mathbf{x}-\mathbf{y}_i)(\mathbf{x}-\mathbf{y}_i)^T))^{-r_2}d\mathbf{x}$ where $\mathbf{A}$ and $\mathbf{B}$ are positive semidefinite matrices, $k$, $r_1$ and $r_2$ are positive scalar constants, $\mathbf{x}$ and $\mathbf{y}_i, i=0,\ldots,n$ are $d$-dimensional vectors.

Any help?

I am stuck in an integral problem:

$$\int_{\mathbb{R}^d}(\det(\mathbf{A}+k(\mathbf{x}-\mathbf{y}_0)(\mathbf{x}-\mathbf{y}_0)^T))^{-r_1}\prod_{i=1}^n(\det(\mathbf{B}+(\mathbf{x}-\mathbf{y}_i)(\mathbf{x}-\mathbf{y}_i)^T))^{-r_2}d\mathbf{x}$$

where $\mathbf{A}$ and $\mathbf{B}$ are positive semidefinite matrices, $k$, $r_1$ and $r_2$ are positive scalar constants, $\mathbf{x}$ and $\mathbf{y}_i, i=0,\ldots,n$ are $d$-dimensional vectors.

Any help?