I am trying to evaluate the following integral:

Integrate[HermiteH[n, Sqrt[a]* x] * Exp[- (c/2)* (x^2 + y^2) + b* x * y - (a * x^2)/2], {x, - Infinity, Infinity}]

but Mathematica just returns the function, e.g. does not evaluate it. It does evaluate for specific values of n, but I need a general result. I have considered using FindSequenceFunction, but I believe that this only works for integer sequences (?)

Is there some way I can get Mathematica to evaluate this integral, or is it impossible to do?

Thanks in advance for any replies.