I am attempting to marginalize a probability density function. But I got stuck on the following integral $$ \int_{-\infty}^\infty\cdots\int_{-\infty}^\infty \frac{\exp(\pmb x^T A\pmb z)} {|\exp(A\pmb z )|_1^{n+|\pmb{x}|_1}} \mathrm dz_1\cdots\mathrm dz_m $$ where $\pmb x, \pmb z \in\mathbb R^n$, $x_i\ge 0$ with large $n,m \in\mathbb N$. $|\cdot|_1$ is the sum of the components e.g. $|\pmb x|_1 = \sum_{i=1}^nx_i$.
$A\in\mathbb R^{n\times n}$ is orthogonal and the first $m$ coloums are also orthogonal to $(1,\dots,1)$.
I already stripped away some constants.
My goal is a fast evaluation of the integral on the remaining dimensions $z_{m+1},\dots,z_n$. Can anybody solve this? If there is no explicit form of the integral, a good approximation would also be very helpful!