In this paper, we consider the observation of $n$ i.i.d. mixed Poisson processes with random intensity having an unknown density $f$ on ${\mathbb R}^+$. For fixed observation time $T$, we propose a nonparametric adaptive strategy to estimate $f$. We use an appropriate Laguerre basis to build adaptive projection estimators. Non-asymptotic upper bounds of the ${\mathbb L}^2$-integrated risk are obtained and a lower bound is provided, which proves the optimality of the estimator. For large $T$, the variance of the previous method increases, therefore we propose another adaptive strategy. The procedures are illustrated on simulated data.
