Computing $\int_{0}^{\infty }x^a\,e^{-bx}\,Q(x)\,dx$, where $Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty }e^{-t^2/2}\,dt$.

Question

I want to know if this integral can be solved:

$$\int_{0}^{\infty }x^a\ e^{-bx}\ Q(x)\ dx\ .$$

where $a,b >0$ are real numbers and $$ Q(x)=\frac{1}{\sqrt{2\pi}}\int_{x}^{\infty }e^{-t^2/2}\,dt = -\dfrac{1}{2}\left( \operatorname{erf}\left(\frac{x}{\sqrt{2}}\right)-1\right)\,. $$

I tried integration by parts, but then the incomplete Gamma function gets involved and it becomes even more complicated. I tried to inverse the integrals but with no success as well. I fear that I am missing over an important function or simplification.

Any Ideas ??