A multi-degree-of-freedom (multi-DOF) wave energy converter (WEC) with hybrid power take-off (PTO) systems may capture more wave energy than a traditional single-DOF WEC, which usually obtains energy from pitch or heave. Compared with the existing WECs with a single PTO, this paper presents the concept design of a multi-DOF WEC with hybrid PTOs named PUWEC, which has prismatic pairs in the hydraulic cylinder and universal pairs in the gear transmission system. Two corresponding PTOs are matched to the motion of the buoy in heave and pitch to enhance wave energy conversion. This work focuses on establishing a time-domain numerical model to simulate the coupled dynamics and power capture performance of the proposed PUWEC. The model tests are carried out in a wave tank to verify the validity of the numerical model. The results show that the numerical model is efficient in predicting the dynamic performance of the PUWEC. Regular and irregular wave conditions are considered in the numerical simulations to explore the performance characteristics of the devices. It is found that the maximum improvement of the captured power of PUWEC is 49.8 % compared with single-DOF PWEC under the regular wave frequency 0.42 Hz, and the capture width ratio of PUWEC is close to 30 % under three wave conditions. In addition, the motion trajectory of the center of gravity of the buoy is analyzed. With the increase in U-PTO damping, the trajectory of the float changes from horizontal bowl-shaped irregular path to inclined regular hollow elliptical orbit. The trajectory coefficient is introduced to provide insights into how the PUWEC improves the energy conversion. The actual working space of the PUWEC is then calculated, which is less than the design working space, indicating that the actual performance of the PUWEC is fully developed. The proposed PUWEC and constructed numerical model which considers the coupled dynamic performance of dual PTOs, can provide valid reference for the design of other multi-DOF WECs with hybrid PTOs.