The dynamics of an initially localized wave packet is studied for the generalized nonlinear Schrödinger equation with a random potential, where the nonlinear term is β |ψ| p ψ and p is arbitrary. Mainly short times for which the numerical calculations can be performed accurately are considered. Long time calculations are presented as well. In particular, the subdiffusive behavior where the average second moment of the wave packet is of the form m2 ≈ tα is computed. Contrary to former heuristic arguments, no evidence for any critical behavior as function of p is found. The properties of α (p) for relatively short times are explored, a scaling property and a maximal value for p ≈ 1 2 are found.