Integral $\iint xy\frac{1}{\sqrt{b}}\exp\left[-\frac{(x-y)^2}{2b}\right]\sqrt{\frac{2}{c}}\exp\left[-\frac{(x+y)^2}{4c}\right].$

Question

I am interested in the following integral $$\int_{-a}^adx\int_{-a}^a\mathop{\mathrm{d}y}xy\frac{1}{\sqrt{b}}\exp\left[-\frac{(x-y)^2}{2b}\right]\sqrt{\frac{2}{c}}\exp\left[-\frac{(x+y)^2}{4c}\right].$$ Does any one know how to evaluate the above integral? Mathematica was not able to give result for this.