Filament eruptions often lead to coronal mass ejections (CMEs), which can affect critical technological systems in space and on the ground when they interact with the geo-magnetosphere in high speeds. Therefore, it is an important issue to investigate the acceleration mechanisms of CMEs in solar/space physics. Based on observations and simulations, the resistive magnetic reconnection and the ideal instability of magnetic flux rope have been proposed to accelerate CMEs. However, it remains elusive whether both of them play a comparable role during a particular eruption. It has been extremely difficult to separate their contributions as they often work in a close time sequence during one fast acceleration phase. Here we report an intriguing filament eruption event, which shows two apparently separated fast acceleration phases and provides us an excellent opportunity to address the issue. Through analyzing the correlations between velocity (acceleration) and soft (hard) X-ray profiles, we suggest that the instability and magnetic reconnection make a major contribution during the first and second fast acceleration phases, respectively. Further, we find that both processes have a comparable contribution to accelerate the filament in this event.