This paper deals with state estimation of nonlinear stochastic dynamic models. In particular, the stochastic integration rule, which provides asymptotically unbiased estimates of the moments of nonlinearly transformed Gaussian random variable, is reviewed together with the recently introduced stochastic integration filter. Then, the respective multi-step prediction and smoothing algorithms are developed in full and efficient square-root form. The stochastic-integration-rule-based algorithms are implemented in Python (within the Stone Soup project) and in MATLAB and numerically evaluated and compared with the well-known unscented Kalman filter. This repository contains the MATLAB implementation.