A multi time delay HIV infection model with adaptive immune responses is proposed, in which both the virus-to-cell infection and the cell-to-cell transmission are considered. The existence of five equilibria and five basic reproduction numbers are calculated. By using the Lyapunov functionals, the sufficient conditions on the global stability of five equilibria are established. Using a time delay $\tau_3$ as a bifurcation parameter, we show that $\tau_3$ may destabilize two equilibria $\widetilde{E}_2$ and $\widetilde{E}_4$ leading to Hopf bifurcation. The results indicate that $\tau_3$ can lead to periodic oscillations in viral load and immune responses, which can reduce the risk of infection. Finally, numerical simulations are carried out to illustrate the corresponding theoretical results, and reveal the effects of different delay parameters on the stability of the equilibria $\widetilde{E}_2$ and $\widetilde{E}_4$.