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.