\[\int_{0}^{\infty}x^{a-1}e^{-sx}\Gamma\left(b,x\right)\,\mathrm{d}x=\frac{%
\Gamma\left(a+b\right)}{a(1+s)^{a+b}}\*F\left(1,a+b;1+a;s/(1+s)\right),\]