Natural fish possess remarkable aquatic abilities, which have motivated the development of flexible robotic fish. However, the complexity of the flexible deformation mechanism poses a significant challenge to analyzing and optimizing the energy cost of these robots. To address this issue, this paper proposes an energy-saving analysis and power cost optimization method for a bionic robotic fish with multiple flexible joints. First, via Lagrange equation and pseudo-rigid body model, a detailed dynamic model is established to theoretically evaluate the swimming performance of the flexible robotic fish. More importantly, by comparing the energy cost characteristics of the robotic fish with multi-flexible joints and multi-active joints, the periodic energy storage of flexible materials is verified to play a key role in reducing the power cost of the robotic fish. Furthermore, a novel framework is established to fully exploit the energy-saving potential of the flexible robotic fish. This framework consists of a two-phase hill climbing method and an improved multi-population genetic algorithm, which work together to effectively optimize the power cost of the robot with speed inequality constraints. Finally, extensive simulation and experiments demonstrate the effectiveness of the proposed method. The obtained results show that the multi-flexible robotic fish exhibits high performance and low energy cost, which holds great promise for the deployment of flexible robotic fish in long-range movement applications.