A double integral with a closed form, generalization

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 $$b \in \mathbb{R}$$.