On an double integral involving Gaussian term

Question

I want to calculate such an integral $$ I=\int_0^{\infty}\int_0^\infty dx\,dy\ x^ny^m\exp(-ax^2-by^2-gxy+c_1x+c_2y), $$ where $a,b,g>0$ are positive and real, $n,m$ are positive integers and $c_1,c_2$ are arbitrary complex numbers. For sure I can firstly integrate over, say $y$, which is $$ \int_0^\infty dy\ y^m\exp(-by^2+(c_2-gx)y)\sim F\left(1+\frac{m}{2},\frac{3}{2},\frac{(c_2-gx)^2}{4b}\right) $$ where $F(\alpha,\gamma,z)$ is the Kummer Hypergeometric function. Now I havo to deal with the integral like this $$ \int_0^ \infty dx\ x^n\exp(-ax^2+c_1x) F\left(\alpha,\gamma,\frac{(c_2-gx)^2}{4b}\right). $$ How can I go on with it?