$\newcommand\s\Sigma$The $(i,j)$-entry of your matrix integral is $$\sum_{k,l}A_{kl}\,EY_iY_jY_kY_l,$$ where $(Y_1,\dots,Y_n)$ is a Gaussian zero-mean random vector with covariance matrix $\Sigma$. In turn, the expectations $EY_iY_jY_kY_l$ can be computed using the Isserlis theorem: $$EY_iY_jY_kY_l=\s_{ij}\s_{kl}+\s_{ik}\s_{jl}+\s_{il}\s_{jk},$$ where $\s_{ij}$ is the $(i,j)$-entry of the matrix $\s$.

So, the $(i,j)$-entry of your matrix integral is $$\sum_{k,l}A_{kl}(\s_{ij}\s_{kl}+\s_{ik}\s_{jl}+\s_{il}\s_{jk}).$$

$\newcommand\s\Sigma$The $(i,j)$-entry of your matrix integral is $$\sum_{k,l}A_{kl}\,EY_iY_jY_kY_l,$$ where $(Y_1,\dots,Y_n)$ is a Gaussian zero-mean random vector with covariance matrix $\Sigma$. In turn, the expectations $EY_iY_jY_kY_l$ can be computed using the Isserlis theorem: $$EY_iY_jY_kY_l=\s_{ij}\s_{kl}+\s_{ik}\s_{jl}+\s_{il}\s_{jk},$$ where $\s_{ij}$ is the $(i,j)$-entry of the matrix $\s$.

So, the $(i,j)$-entry of your matrix integral is $$\sum_{k,l}A_{kl}(\s_{ij}\s_{kl}+\s_{ik}\s_{jl}+\s_{il}\s_{jk}).$$

So, your matrix integral is $$(\text{tr}(A\s))\s+\s A\s+\s A^\top\s.$$

$\newcommand\s\Sigma$The $(i,j)$-entry of your matrix integral is $$\sum_{k,l}A_{k,l}\,EY_iY_jY_kY_l,$$ where $(Y_1,\dots,Y_n)$ is a Gaussian zero-mean random vector with covariance matrix $\Sigma$. In turn, the expectations $EY_iY_jY_kY_l$ can be computed using the Isserlis theorem: $$EY_iY_jY_kY_l=\s_{ij}\s_{kl}+\s_{ik}\s_{jl}+\s_{il}\s_{jk}.$$

$\newcommand\s\Sigma$The $(i,j)$-entry of your matrix integral is $$\sum_{k,l}A_{kl}\,EY_iY_jY_kY_l,$$ where $(Y_1,\dots,Y_n)$ is a Gaussian zero-mean random vector with covariance matrix $\Sigma$. In turn, the expectations $EY_iY_jY_kY_l$ can be computed using the Isserlis theorem: $$EY_iY_jY_kY_l=\s_{ij}\s_{kl}+\s_{ik}\s_{jl}+\s_{il}\s_{jk},$$ where $\s_{ij}$ is the $(i,j)$-entry of the matrix $\s$.

So, the $(i,j)$-entry of your matrix integral is $$\sum_{k,l}A_{kl}(\s_{ij}\s_{kl}+\s_{ik}\s_{jl}+\s_{il}\s_{jk}).$$