Integrals Involving Exponential and Determinant

Question

Suppose $a_{ij}=a_{ij}(x_1, x_2, \ldots, x_n)$ are symmetric functions in the variables $x_1, x_2, \ldots, x_n$. I'm wondering if there is a general method or approach for evaluating integrals of the form $$\int_{\mathbb{R}^n} e^{-c\det(a_{ij})}\Delta^2\,dx_1dx_2\cdots dx_n,$$ where $c>0$ and $\Delta$ denotes the Vandermonde determinant. This type of integral appears in the $\mbox{GUE}(n)$ measure on $n\times n$ complex Hermitian matrices. In this instance, one would have $$A=(a_{ij})=\begin{bmatrix} 0 & x_1^2+x_2^2+\cdots +x_n^2\\ -1 & 0\end{bmatrix}.$$