Folded n-dimensional gaussian integral

Question

I am dealing with the following integral: \begin{equation} \int_0^{\infty} dx_1 \ldots \int_0^{\infty} dx_n \quad e^{-J^T x - x^T A x} \end{equation} Is there any general analytical solution which gives the above integral in terms of the determinant of $A$, $J$ and (I guess) the error function?

Thanks!

Answer 1

Update: I didn't notice the domain was different. Please ignore this answer.

This is standard in quantum field theory. Take a look here or any book on the topic.