[Community Question] Calculus: Integrate $xe^{-bx/d}\mathrm{erfc}(ax+c)$

One of our user asked:

I want to calculate and evaluate the following integral: $$\frac{B}{2 D}\int_{0}^{\infty} xe^{\frac{-Bx}{D}} erfc(\frac{x+x_{0}-Bt}{2\sqrt{Dt}})$$ My idea was to integrate by parts by setting: $$u= erfc(\frac{x+x_{0}-Bt}{2\sqrt{Dt}}), du=-\frac{1}{\sqrt{Dt}} e^{-(\frac{x+x_{0}-Bt}{2\sqrt{Dt}})^2}$$ $$dv=xe^{-\frac{Bx}{D}},v=e^{-\frac{Bx}{D}}(\frac{Bx}{D}x+1)*\frac{D^2}{B^2}$$

I have calculated further and got some results, but I am not sure if my thinking is right, or even if there is some easier or more efficient way to do this analytically or numerically,(for example by expanding the error function as infinite series). Any tips would be appreciated.

And of course, erfc is the complementary error function with: $$erfc(\frac{x+x_{0}-Bt}{2\sqrt{Dt}})=\frac{2}{\sqrt{\pi}}\int_{\frac{x+x_{0}-Bt}{2\sqrt{Dt}}}^{\infty} e^{-z^2} dz $$