Executing agile quadrotor maneuvers with cable-suspended payloads is a challenging problem and complications induced by the dynamics typically require trajectory optimization. State-of-the-art approaches often need significant computation time and complex parameter tuning. We present a novel dynamical model and a fast trajectory optimization algorithm for quadrotors with a cable-suspended payload. Our first contribution is a new formulation of the suspended payload behavior, modeled as a link attached to the quadrotor with a combination of two revolute joints and a prismatic joint, all being passive. Differently from state of the art, we do not require the use of hybrid modes depending on the cable tension. Our second contribution is a fast trajectory optimization technique for the aforementioned system. Our model enables us to pose the trajectory optimization problem as a Mathematical Program with Complementarity Constraints (MPCC). Desired behaviors of the system (e.g., obstacle avoidance) can easily be formulated within this framework. We show that our approach outperforms the state of the art in terms of computation speed and guarantees feasibility of the trajectory with respect to both the system dynamics and control input saturation, while utilizing far fewer tuning parameters. We experimentally validate our approach on a real quadrotor showing that our method generalizes to a variety of tasks, such as flying through desired waypoints while avoiding obstacles, or throwing the payload toward a desired target. To the best of our knowledge, this is the first time that three-dimensional, agile maneuvers exploiting the system dynamics have been achieved on quadrotors with a cable-suspended payload.