We propose a new exact solution algorithm for closed multiclass product-form queueing networks that is several orders of magnitude faster and less memory consuming than established methods for multiclass models, such as the Mean Value Analysis (MVA) algorithm. The technique generalizes the recently proposed Method of Moments (MoM) which, differently from MVA, recursively computes {higher-order} moments of queue lengths instead of mean values. The main contribution of this paper is to show that the information used in the MoM recursion can be increased by considering multiple recursive branches that evaluate models with fewer queues. This reformulation allows to define a simpler matrix difference equation for computing normalizing constants which leads to large computational savings with respect to the MoM recursion. Computational analysis shows many cases where the proposed algorithm is between 1,000 and 10,000 times faster and less memory consuming than MoM, thus extending the range of multiclass models where exact solutions are feasible.