Probabilistic inference in high-dimensional continuous or hybrid domains poses significant challenges, commonly addressed through discretization, sampling, or reliance on parametric assumptions. The drawbacks of these methods are well-known: inaccuracy, slow computational speeds or overly constrained models. This paper introduces a novel general inference algorithm designed for Bayesian networks featuring both discrete and continuous variables. The algorithm avoids the discretization of continuous densities into histograms by employing quadrature rules to compute continuous integrals and avoids the use of a parametric model by using orthogonal polynomials to represent the posterior density. Additionally, it preserves the computational efficiency of classical sum-product algorithms by using an auxiliary discrete Bayesian networks appropriately constructed to make continuous inference. Numerous experiments are conducted using either the conditional linear Gaussian model as a benchmark, or non-Gaussian models for greater generality. Our algorithm demonstrates significant improvements both in speed and accuracy when compared with existing methods.