The trajectory optimization of a spacecraft is considered in the gravitational effects of the Moon, Earth, and Sun in the paper. Imposing practical constraints of maximum thrust, fuel budget, and flight time generates a constrained, non-autonomous, nonlinear optimal control problem. Severe constraints on the fuel budget combined with high accuracy demands on the endpoint conditions necessitates a high-accuracy solution to the trajectory optimization problem. The problem is first solved using the standard Legendre pseudospectral method. The optimality of the solution is verified by an application of the Covector Mapping Principle. It is shown that the thrust structure consists of three finite burns with nearly linear steering-angle time histories. A singular arc is detected and is interpreted as a singular plane-change maneuver. The Bellman pseudospectral method is then employed to improve the accuracy of the solution.