To address the trajectory tracking issue for quadrotors working under the influence of external disturbances, a control scheme is designed for position and attitude trajectory tracking. Firstly, the quadrotor dynamic model is derived for control design. Secondly, adaptive integral backstepping control (AIBS) is employed in the position loop in which adaptive estimation is used to estimate the upper bounds of disturbances. Subsequently, a new adaptive backstepping fast nonsingular integral terminal sliding mode control (ABFNITSM) is proposed to follow desired Euler angles. The intro-duction of fast nonsingular terminal sliding mode and integral element is to realize fast convergence and accurate tracking. Dead zone technique is applied to minimize estimation errors. Besides, a saturation function is utilized to eliminate chattering phenomenon. Finally, simulation experiments are conducted under the Simulink environment. The proposed control is verified by comparing it with traditional integral terminal sliding mode control (ITSM) and integral backstepping control (IBS).