This paper presents a new traffic engineering technique for dynamic constrained multicast routing, where the routing request of traffic arrives one-by-one. The objective we adopted is to minimize the maximum of link utilization. Although this traffic engineering is useful to relax the most heavily congested link in the Internet backbone, the total network resources, i.e. sum of link bandwidth consumed, could be wasted when the acquired path is larger (in terms of number of hops) than the conventional shortest path. Accordingly we find a multicast tree for routing request that satisfies the hop-count constraint. We formulate this problem as a mixed-integer programming problem and propose a new heuristic algorithm to find a multicast tree for multicast routing request. The presented heuristic algorithm uses the link-state information, i.e. link utilization, for multicast tree selection and is amenable to distributed implementation. The extensive simulation results show that the proposed traffic engineering technique and heuristic algorithm efficiently minimize the maximum of link utilization better than the shortest path.