The development of the energy internet promotes the efficiency, economy and security of the energy system. In the Integrated Electrical and Natural Gas Network (IEGN), linepack reserves not only secure the safe operation of the natural gas network, but also could be employed by gas turbines to provide fast-response flexibility. It plays an increasingly significant role in balancing the electrical network with renewable energy sources and multi-energy utilizations. Besides, the linepack reserve also guards the coordination between networks, and the normal function of multi-energy utilizations, including combined cooling, heat and power, aka trigeneration plants. To promote the safety and coordination of the IEGN, this paper novelly integrates the linepack reserve into the dispatch process and proposes a multi-objective optimal dispatch model for the IEGN with trigeneration plants to simultaneously promote its economy, safety and efficiency. NSGAII is employed to solve it. Finally, case studies demonstrate its effectiveness. This work is supported by Fundamental Research Funds for the Central Universities (No. 2016XS13).