A complicated integral / a complicated Laplace transform involving the error function

For some reason I am interested in solving a complicated integral, which is $$\int_0^\infty (x+1)erf\left(\frac{-c_1+c_2x}{b}\right)e^{-c_3x^2-c_4x}dx,$$ where all $$c_i$$ are positive real numbers and $$erf$$ denotes the error function. Equivalently, one can also talk about the Laplace transform of $$(x+1)erf\left(\frac{-c_1+c_2x}{b}\right)e^{-c_3x^2}$$ at $$c_4$$.

While I know that there are many tables for solving stuff like that, I haven't found anything that solves the above expression. If you have any hints, I'd be very happy to read them.

• there is little hope for a closed form expression; if some parameters are small or large you could make progress, but for arbitrary parameters you'll want to evaluate this numerically. – Carlo Beenakker Nov 19 at 22:20