This research deals with the computation of optimal trajectories considering state and input constraints for linear and nonlinear systems that admit a polynomial representation through differential flatness. Based on a polynomial spline parameterization of the flat output an optimization problem in terms of the B-spline coefficients is derived that guarantees constraint satisfaction over the entire time horizon whereas classical approaches in the literature only impose the constraints on a finite time grid. As the proposed constraints are only sufficient conditions, a novel method is presented that effectively reduces their conservatism. Two numerical examples, a linear benchmark tracking problem and an optimal quadrotor maneuver, illustrate the efficiency and practicality of the presented method.