A first integral for a class of time-dependent anharmonic oscillators with multiple anharmonicities

The integrability of the equation q̈=f(t)q2+g(t)q3+h(t)q4+j(t)q5 is considered. Particular cases of this equation arise in the study of charged plasma in an axially symmetric magnetic field and in shear-free spherically symmetric gravitational fields in general relativity. The above equation with only a quadratic term arises in the study of shear-free fluids, in general [A. Krasinski, J. Math. Phys. 30, 433 (1989)]. The equation with a cubic term is applicable when there is an electromagnetic field. In special cases we reduce the solution to a quadrature that has solutions in terms of elliptic integrals. A Lie point symmetry analysis is performed and the different cases that arise are considered for the existence of a symmetry.