Weighted model integration (WMI) is a very appealing framework for probabilistic inference: it allows to express the complex dependencies of real-world scenarios where variables are both continuous and discrete, via the language of Satisfiability Modulo Theories (SMT), as well as to compute probabilistic queries with complex logical and arithmetic constraints. Yet, existing WMI solvers are not ready to scale to these problems. They either ignore the intrinsic dependency structure of the problem at all, or they are limited to too restrictive structures. To narrow this gap, we derive a factorized formalism of WMI enabling us to devise a scalable WMI solver based on message passing, MP-WMI. Namely MP-WMI is the first WMI solver which allows to: 1) perform exact inference on the full class of tree-structured WMI problems; 2) compute all the marginal densities in linear time; 3) amortize inference for any query conforming to the problem structure. Experimental results show that our solver dramatically outperforms the existing WMI solvers on a large set of benchmarks.