$$\int \exp\left(-\frac{1}{2}x^TAx+B^Tx\right)dx$$ where A is a real and positive-definite, i.e. $x^TAx>0$, but not necessarily symmetric. $B$ and $x$ are $n\times 1$ column vectors with real components.
When A is symmetric and positive definite, the result is the well known $$\int\exp\left(-\frac{1}{2}x^TAx+B^Tx\right)=\sqrt\frac{(2\pi)^n}{\det\left(A\right)}\exp\left(B^TA^{-1}B\right)$$ I have also found that when $A$ is real and positive definite but not necessarily symmetric, which is worked here: (https://pdfs.semanticscholar.org/13aa/494c312d57a92d72e408bc4f3503edaac8eb.pdf) $$\int\exp\left(-\frac{1}{2}x^TAx\right)=\sqrt\frac{(\pi)^n}{det\left(\frac{1}{2}\left(A+A^T\right)\right)}$$ I am thinking if it's possible to generalize the above result with a linear term $B^Tx$ for a non-symmetric $A$.