Summary: A known one-dimensional, undamped, anharmonic, unforced oscillator whose restoring force is a displacement’s odd polynomial function, is exactly solved via the Gauss and Appell hypergeometric functions, revealing a new fully integrable nonlinear system. Our \(t=t(x)\) equation – and its correspondent \(x=x(t)\) obtained via the Lagrange reversion approach – can then added to the (not rich) collection of highly nonlinear oscillating systems integrable in closed form. Finally, the hypergeometric formula linking the period \(T\) to the initial motion amplitude \(a\) is then assumed as a benchmark for ranking the approximate values of the relevant literature.