This paper consists in introducing an option price expansion for model combining local and stochastic volatility {with tight error estimates}. The local volatility part is considered as general but has to satisfy some growth and boundedness assumptions. For the stochastic part, we choose a square root process, which is usually used for modelling the behaviour of the variance process. In the particular case of Call options, we also provide expansions of the Black-Scholes implied volatility which allow to obtain very simple and rapid formulas in comparison to the Monte Carlo approach while maintaining a very competitive accuracy.