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Internet traffic is a mixture of unicast and multicast flows. Integrated schedulers capable of dealing with both traffic types have been designed mainly for Input Queued (IQ) buffer-less crossbar switches. Combined Input and crossbar queued (CICQ) switches, on the other hand, are known to have better performance than their buffer-less predecessors due to their potential in simplifying the scheduling and improving the switching performance. The design of integrated schedulers in CICQ switches has thus far been neglected. In this paper, we propose a novel CICQ architecture that supports both unicast and multicast traffic along with its appropriate scheduling. In particular, we propose an integrated round-robin-based scheduler that efficiently services both unicast and multicast traffic simultaneously. Our scheme, named multicast and unicast round robin scheduling (MURS), has been shown to outperform all existing schemes under various traffic patterns. Simulation results suggested that we can trade the size of the internal buffers for the number of input multicast queues. We further propose a hardware implementation of our algorithm for a 16 times 16 buffered crossbar switch. The implementation results suggest that MURS can run at 20 Gbps line rate and a clock cycle time of 2.8 ns, reaching an aggregate switching bandwidth of 320 Gbps.