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In this paper, a quality-of-service (QoS) multicast routing protocol, i.e., the core-based tree based on heuristic genetic algorithms (GAs), has been implemented and applied over a high-altitude platform (HAP)-satellite platform. The proposed multicast routing algorithm is called the constrained cost-bandwidth-delay GA (CCBD-GA). To achieve a better optimization of the multicast tree cost, a new algorithm called HULK-GA, which is based on the GA and on a proposed broadcast metric, has been developed. Finally, an algorithm called hybrid cost-bandwidth-delay GA has been proposed, taking into account both CCDB-GA and HULK-GA characteristics to obtain an overall algorithm that can consider QoS routing constraints and minimize the overall cost per link of the considered multicast tree. The joint bandwidth-delay metrics can be very useful in hybrid platforms such as the platform considered, because it is possible to take advantage of the single characteristics of the satellite and HAP segments. The HAP segment offers low propagation delay, permitting QoS constraints based on maximum end-to-end delay to be met. The satellite segment, instead, offers a larger footprint but higher propagation delay. The joint bandwidth-delay metric permits the traffic load to be balanced, respecting both QoS constraints.