In multirate multicasting, different users (receivers) within the same multicast group can receive service at different rates, depending on the user requirements and the network congestion level. Compared with unirate multicasting, this provides more flexibility to the user and allows more efficient usage of the network resources. In this paper, we address the rate control problem for multirate multicast sessions, with the objective of maximizing the total receiver utility. This aggregate utility maximization problem not only takes into account the heterogeneity in user requirements, but also provides a unified framework for diverse fairness objectives. We propose an algorithm for this problem and show, through analysis and simulation, that it converges to the optimal rates. In spite of the nonseparability of the problem, the solution that we develop is completely decentralized, scalable and does not require the network to know the receiver utilities. The algorithm requires very simple computations both for the user and the network, and also has very low overhead of network congestion feedback.