I have encountered a double integral with three parameters which has the following form:
$$I(a,b,c)= \int_{-\infty}^\infty \int_{-c}^\infty \frac{e^{-a (x^2+b x+y)} \mathrm d y \mathrm d x}{\sqrt{(x^2+y)^2+4 c x^2}}$$
I we set $b=0$, then use a lot of different substitutions, we can get the closed form:
$$I(a,0,c)= \int_{-\infty}^\infty \int_{-c}^\infty \frac{e^{-a (x^2+y)} \mathrm d y \mathrm d x}{\sqrt{(x^2+y)^2+4 c x^2}}= \frac{\pi}{\sqrt{a}} \frac{\operatorname{erfi} (\sqrt{a c})}{\sqrt{a c}}$$
This value checks out numerically and if needed, I can provide the full derivation.
However, I'm still interested in the more general case with $b \neq 0$. The problem I have from the start: the function is no longer even in $x$ which I used in the first step of my solution for $b=0$.
Is there a closed form for $I(a,b,c)$? With $b \in \mathbb{R}$.