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In this article we prove that all real quadratic differential systems dx dt = p(x, y), dy dt = q(x, y) with gcd(p, q) = 1, having invariant lines of total multiplicity at least five and a finite set of singularities at infinity, are Darboux integrable having integrating factors whose inverses are polynomials over R. We also classify these systems under two equivalence relations: 1) topological equivalence and 2) equivalence of their associated cubic projective differential equations when cubic projective differential equations are acted upon by the group PGL(2,R). For each one of the 28 topological classes obtained we give necessary and sufficient conditions for such a system with invariant lines to belong to this class, in terms of its coefficients in R12. Résumé Dans cet article nous prouvons que tous les systèmes différentiels réels dx dt = p(x, y), dy dt = q(x, y), gcd(p, q) = 1, ayant des droites invariantes de multiplicité totale au moins cinq, sont Darboux intégrable possédant des facteurs intégrant dont les inverses sont des polynomes sur R. Nous classifions ces systèmes par rapport à deux relations d’équivalence : 1) l’équivalence topologique et 2) l’équivalence de leurs équations différentielles cubiques projectives sous l’action du groupe PGL(2,R). Pour chaqune des 28 classes topologiques obtenues nous donnons des conditions nécessaires et suffisantes pour qu’un tel système ayant des droites invariantes appartienne à cette classe, en terms de ses coefficients dans R12.

Integrals and Phase Portraits of Planar Quadratic Differential Systems With Invariant Lines of at Least Five Total Multiplicity