Good estimate for ratio $\frac{\gamma(d + s, x)}{\gamma(d,x)}$, where $\gamma$ is the lower incomplete gamma function.

Question

Let $\gamma(d,x) := \int_{0}^xe^{-t}t^{d-1}dt$ (with $d,x > 0$) be the lower incomplete gamma function.

Question. For $d,s,x > 0$ with $d \gg s$, what is a good estimate for the ratio $$ \frac{\gamma(d + s, x)}{\gamma(d,x)}. $$

I'm hoping some subtle application of Stirling's inequality is doable here.