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 the cubic projective differential equations are acted upon by the group PGL (3, R). For each one of the 28 topological classes obtained, we give necessary and sufficient conditions for a quadratic system to belong to this class, in terms of its coefficients in R ¹².