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We provide the necessary and sufficient conditions of Liouvillianintegrability for Li\'{e}nard differential systems describing nonlinearoscillators with a polynomial damping and a polynomial restoring force. Weprove that Li\'{e}nard differential systems are not Darboux integrableexcluding subfamilies with certain restrictions on the degrees of thepolynomials arising in the systems. We demonstrate that if the degree of apolynomial responsible for the restoring force is greater than the degree of apolynomial producing the damping, then a generic Li\'{e}nard differentialsystem is not Liouvillian integrable with the exception of linear Li\'{e}nardsystems. However, for any fixed degrees of the polynomials describing thedamping and the restoring force we present subfamilies possessing Liouvillianfirst integrals. As a by-product of our results, we find a number of novelLiouvillian integrable subfamilies. In addition, we study the existence ofnon-autonomous Darboux first integrals and non-autonomous Jacobi lastmultipliers with a time-dependent exponential factor.

Integrability and solvability of polynomial Li\'{e}nard differential systems