A complete analytical solution of the mathematical model describing the nonstationary nucleation and growth of crystals in supercooled melts and supersaturated solutions is obtained, taking into account the nucleation kinetics of Meiers and Weber-Volmer-Frenkel-Zeldovich, and also the power law dependence the growth rate of crystallites from supercooling or supersaturation of the system. The solution of the problem is based on the application of the saddle point method to calculate the Laplace-type integral. It is shown that for a correct description of the evolution of a metastable system, it is necessary to take into account both the fundamental term of the saddle-point method and the next four terms of the asymptotic