The integrated radar and communication system is promising in the next generation wireless communication networks. However, its performance is confined by the limited energy. In order to overcome it, a wireless powered integrated radar and communication system is proposed. An energy minimization problem is formulated subject to constraints on the radar and communication performances. The energy beamforming and radar-communication waveform are jointly optimized to minimize the consumption energy. The challenging non-convex problem is solved by using semidefinite relaxation and auxiliary variable methods. It is proved that the optimal solution can be obtained. Simulation results demonstrate that our proposed optimal design outperforms the benchmark scheme.