This paper proposes a new analytical probabilistic power flow (PPF) approach for power systems with high penetration of distributed energy resources. The approach solves probability distributions of system variables about operating conditions. Unlike existing analytical PPF algorithms in literature, this new approach preserves nonlinearities of ac power flow equations and retain more accurate tail effects of the probability distributions. The approach first employs a multidimensional holomorphic embedding method to obtain an analytical nonlinear ac power flow solution for concerned outputs such as bus voltages and line flows. The embedded symbolic variables in the analytical solution are the inputs such as power injections. Then, the approach derives cumulants of the outputs by a generalized cumulant method, and recovers their distributions by Gram-Charlier expansions. This PPF approach can accept both parametric and nonparametric distributions of random inputs and their covariances. Case studies on the IEEE 30-bus system validate the effectiveness of the proposed approach.