Solve the numerical value of this integral $\int_0^\infty t^{a-1}e^{-t} \Gamma(b,t)dt$

Question

I need to compute the numerical value of this integral, hundred thousand of times, for a typical dataset. How can I get a good approximation.

$$ \int_0^\infty t^{a-1}e^{-t} \Gamma(b,t)dt $$ where a and b are positive integer and $\Gamma(b,t)=\int_t^\infty \lambda^{b-1}e^{-\lambda}d\lambda$

Answer 1

Heres how I tried to derive a solution for your integral, of course more rigor could be applied to say why the steps are valid but I think this is sufficient