•Closed-form formula for conditional moments of generalized NLD-CEV process.•Closed-form formulas for conditional and unconditional moments of a nonlinear drift CEV process are presented in simplified form.•Well-known instances deduced by the nonlinear drift CEV process are the Cox-Ingersoll-Ross and inverse Feller processes or 3/2-stochastic volatility model.•Sufficient conditions of existence and uniqueness of a positive pathwise strong solution for time-dependent parameter functions are provided. This paper studied a generalized case of the constant elasticity of variance diffusion (CEV) process whereas the drift term is substantially nonlinear in the short rate. Well-known instances deduced by this process are the extended Cox–Ingersoll–Ross (ECIR) process and the extended inverse Feller (EIF) process or 3/2-stochastic volatility model (SVM). We found particular sufficient conditions of existence and uniqueness of a positive pathwise strong solution for time-dependent parameter functions, and obtained closed-form formulas for conditional moments based on Feynman–Kac theorem. The accuracy and validity of the formulas were further investigated based on Monte Carlo simulations.