Gamma-Poisson Mixing Produces Negative Binomial Counts

A latent intensity $\Lambda$ is $\mathrm{Gamma}(\alpha,\beta)$ with shape $\alpha$ and rate $\beta$. Conditional on $\Lambda$, the count $N$ is $\mathrm{Poisson}(\Lambda)$. Use MGFs to identify the unconditional distribution of $N$.