This article presents a mathematical model of vapor bubble growth in a superheated liquid, which simultaneously takes into account both dynamic and thermal effects and includes the well-known classical equations, the momentum equation and the heat equation, written to take into account the process of liquid evaporation. An approximate semi-analytical solution of the problem is found, its construction based on the existence of a quasi-stationary state for the bubble growth process. This makes it possible to reduce the original moving boundary value problem to a system of ordinary differential equations of the first order. The solution obtained is valid at all stages of the process and for a wide range of system parameters. It is shown that at large times the solution becomes self-similar and in limiting cases it agrees with the known solutions of other authors.