  We provide the necessary and sufficient conditions of Liouvillian
integrability for Li\'{e}nard differential systems describing nonlinear
oscillators with a polynomial damping and a polynomial restoring force. We
prove that Li\'{e}nard differential systems are not Darboux integrable
excluding subfamilies with certain restrictions on the degrees of the
polynomials arising in the systems. We demonstrate that if the degree of a
polynomial responsible for the restoring force is greater than the degree of a
polynomial producing the damping, then a generic Li\'{e}nard differential
system is not Liouvillian integrable with the exception of linear Li\'{e}nard
systems. However, for any fixed degrees of the polynomials describing the
damping and the restoring force we present subfamilies possessing Liouvillian
first integrals. As a by-product of our results, we find a number of novel
Liouvillian integrable subfamilies. In addition, we study the existence of
non-autonomous Darboux first integrals and non-autonomous Jacobi last
multipliers with a time-dependent exponential factor.
