Can we find the sufficient and necessary condition of $a$, $b$ and $c$ $\in\mathbb R_+$ such that the following integration is integrable? $$ I_1\equiv\int \frac{1}{|\mathbf{\Sigma}|^a|\mathbf{\Xi}|^b|\mathbf{L}\mathbf{\Sigma}\mathbf{L}'+\mathbf{\Xi}|^c} \mathrm{etr}\left\{-\frac{1}{2} (\mathbf{L}\mathbf{\Sigma}\mathbf{L}'+\mathbf{\Xi})^{-1}\mathbf{X}\right\} d\,\mathbf{\Sigma}~ d\,\mathbf{\Xi}\ , $$ where $\mathrm{etr}(\mathbf{A})$ means $\exp\{\mathrm tr(\mathbf{A})\}$. $\mathbf{\Sigma}=diag\{\sigma_1,\cdots,\sigma_k\}$, $\mathbf{\Xi}=diag\{\xi_1,\cdots,\xi_k\}$, $(\sigma_1,\cdots,\sigma_k)'\in\mathbb R^k_+$, $(\xi_1,\cdots,\xi_k)'\in\mathbb R^k_+$.

$\bf{X}$ is a nonnegative definite $k\times k$ matrix, $\bf{L}$ is a invertibe $k\times k$ matrix.

Can we find the sufficient and necessary condition of $a$, $b$ and $c$ $\in\mathbb R_+$ such that the following integration is integrable? $$ I_1\equiv\int \frac{1}{|\Sigma|^a|\Xi|^b|\mathrm{L}\Sigma\mathrm{L}'+\Xi|^c} \mathrm{etr}\left\{-\frac{1}{2} (\mathrm{L}\Sigma\mathrm{L}'+\Xi)^{-1}\mathrm{X}\right\} d\,\Sigma~ d\,\Xi\ , $$

where etr$(\mathrm{A})$ means exp$\{\mathrm tr(\mathrm{A})\}$. $\Sigma=diag\{\sigma_1,\cdots,\sigma_k\}$, $\Xi=diag\{\xi_1,\cdots,\xi_k\}$, $(\sigma_1,\cdots,\sigma_k)'\in\mathbb R^k_+$, $(\xi_1,\cdots,\xi_k)'\in\mathbb R^k_+$.

$\mathrm{X}$ is a nonnegative definite $k\times k$ matrix, $\mathrm{L}$ is a invertibe $k\times k$ matrix.

Can we find the sufficient and necessary condition of $a$, $b$ and $c$ $\in\mathbb R_+$ such that the following integration is integrable? $$ I_1\equiv\int \frac{1}{|\mathbf{\Sigma}|^a|\mathbf{\Xi}|^b|\mathbf{L}\mathbf{\Sigma}\mathbf{L}'+\mathbf{\Xi}|^c} \mathrm{etr}\left\{-\frac{1}{2} (\mathbf{L}\mathbf{\Sigma}\mathbf{L}'+\mathbf{\Xi})^{-1}\mathbf{X}\right\} d\,\mathbf{\Sigma}~ d\,\mathbf{\Xi}\ , $$ where $\mathrm{etr}(\mathbf{A})$ means $\exp\{\mathrm tr(\mathbf{A})\}$. $\mathbf{\Sigma}=diag\{\sigma_1,\cdots,\sigma_k\}$, $\mathbf{\Xi}=diag\{\xi_1,\cdots,\xi_k\}$, $(\sigma_1,\cdots,\sigma_k)'\in\mathbb R^k_+$, $(\xi_1,\cdots,\xi_k)'\in\mathbb R^k_+$.

$\bf{L}$ is a invertibe $k\times k$ matrix, $\bf{X}$ is a nonnegative definite $k\times k$ matrix.

Can we find the sufficient and necessary condition of $a$, $b$ and $c$ $\in\mathbb R_+$ such that the following integration is integrable? $$ I_1\equiv\int \frac{1}{|\mathbf{\Sigma}|^a|\mathbf{\Xi}|^b|\mathbf{L}\mathbf{\Sigma}\mathbf{L}'+\mathbf{\Xi}|^c} \mathrm{etr}\left\{-\frac{1}{2} (\mathbf{L}\mathbf{\Sigma}\mathbf{L}'+\mathbf{\Xi})^{-1}\mathbf{X}\right\} d\,\mathbf{\Sigma}~ d\,\mathbf{\Xi}\ , $$ where $\mathrm{etr}(\mathbf{A})$ means $\exp\{\mathrm tr(\mathbf{A})\}$. $\mathbf{\Sigma}=diag\{\sigma_1,\cdots,\sigma_k\}$, $\mathbf{\Xi}=diag\{\xi_1,\cdots,\xi_k\}$, $(\sigma_1,\cdots,\sigma_k)'\in\mathbb R^k_+$, $(\xi_1,\cdots,\xi_k)'\in\mathbb R^k_+$.

$\bf{X}$ is a nonnegative definite $k\times k$ matrix, $\bf{L}$ is a invertibe $k\times k$ matrix.