Two methods are used to develop nondivergent, compact, sum-over-states formulas for dynamic hyperpolarizabilities up to fifth order. Expressions for both atoms and molecules are given; those at fourth and fifth order for the first time. The fifth-order formula is related to such nonlinear optical phenomena as stimulated hyper-Raman scattering and the self-induced intensity-dependent refractive index. Questions of time-reversal invariance are also discussed. Importantly, the derivations explicitly take into account secular divergences in a simple and clear way.