This paper addresses the nonlinear control problem of balancing an inverted pendulum on an underactuated unmanned aerial vehicle. In view of the system's slow and quick transients, a convenient error transformation is employed to render the attitude subsystem less sensitive to translational errors. Linearizations are not involved, making this controller suitable beyond the limited scope of trim maneuvers. Furthermore, the controller features an adaptive bounded law that is used to compensate for unknown exogenous perturbations. The Lyapunov-based control design is rooted in an integral backstepping process, wherein the origin of the closed-loop total system error is shown to be almost globally asymptotically stable. Simulation results are presented to validate our strategy, and experimental results are also provided to demonstrate its accuracy and robustness.