After a fair bit of effort, I managed to prove that
$$\int_0^\infty t^\alpha \exp(-t) L_n^{\alpha+1}(t)\mathrm dt=\Gamma(\alpha+1)$$
where $L_n^\alpha (t)$ is a generalized Laguerre polynomial, with the usual restriction of $\Re \alpha > -1$. My proof was rather complicated (involving lots of integration by parts); I'm wondering if there may be a simpler way of proving this proposition.