The valuation of European options under the Heston model (or any other stochastic volatility model where the characteristic function is analytically known) involves the computation of a Fourier transform type of numerical integration. This paper describes how adaptive Filon and adaptive Flinn quadratures may be used to calculate this integral efficiently in accordance with a level of accuracy defined by the user. We then compare the accuracy and the performance of our quadratures with that of others commonly used for this problem, such as the optimal alpha method applied by Lord and Kahl. Finally, the paper concludes with a concrete case of calibration of the model on different sets of market data.