This work investigates the energy shaping control of a class of underactuated mechanical systems with first-order actuator dynamics and subject to both matched and unmatched constant additive disturbances. To this end, a new nonlinear control law which includes two independent integral actions is presented. The controller design is outlined for systems with first-order actuator dynamics, and also for systems with direct actuation. The effectiveness of the proposed approach is demonstrated with numerical simulations on an inertia wheel pendulum and on a ball-on-beam system, both actuated by electric DC motors and subject to constant disturbances.