The generalized Liénard polynomial differential systems x′=yx′=y, y′=−g(x)−f(x)yy′=−g(x)−f(x)y with degg=degf+1 are not Liouvillian integrable

We prove the nonexistence of Liouvillian first integrals for the generalized Liénard polynomial differential systems of the form x′=yx′=y, y′=−g(x)−f(x)yy′=−g(x)−f(x)y, where g(x)g(x) and f(x)f(x) are arbitrary polynomials such that degg=degf+1.