What is the solution to the following integral?
$$\int_{x_1}^{x_2} \int_{y_1}^{y_2} \exp\left(i \pi \left(a x^2 -2bxy + cy^2\right)\right) \, dy \,dx$$
I have tried evaluating it in mathematica with Integrate[Exp[I*Pi*(a*x^2 - 2*b*x*y + c*y^2)], x, y] and it doesn't seem to do it, returning a partially evaluated result
-((-1)^(3/4)*Integrate[E^((I*(-b^2 + a*c)*Pi*x^2)/c)*Erfi[((-1)^(1/4)*Sqrt[Pi]*(-(b*x) + c*y))/Sqrt[c]], x])/(2*Sqrt[c])
$$-\frac{(-1)^{3/4} \int e^{\frac{i \pi x^2 \left(a c-b^2\right)}{c}} \operatorname{erfi}\left(\frac{\sqrt[4]{-1} \sqrt{\pi } (c y-b x)}{\sqrt{c}}\right) \, dx}{2 \sqrt{c}}$$
I have also tried looking in the G&R table of integrals and couldn't find anything that would help me solve it. Is this integral even solvable?
I am aware that this integral has a well known solution as a product of two error functions if the cross term is zero ($b = 0$). I have considered doing a coordinate transformation to cancel out the cross term, however the integration limits then no longer lie along the coordinate axes and hence become parametric, of which I am even less sure of how to deal with. Taking the integration limits to instead lie along the transformed axes doesn't give the solution to the original problem.