The system of differential equations under consideration is \[ x' = x(Cy+z),\quad y' = y(Az+x),\quad z' = z(Bx+y), \] where \(A\), \(B\), and \(C\) are nonzero complex constants. This system is of broad interest because it typically serves as a normal form for ``factored quadratic systems,'' quadratic homogeneous systems such that \(\alpha\) factors out of \(\alpha'\) for each \(\alpha \in \{x, y, z \}\). A first integral for the system is a function that is constant on trajectories. The chief result of this paper is a characterization of those triples \((A, B, C)\) for which the system of ordinary differential equations has a Liouvillian first integral. Specific first integrals, or Darboux polynomials, are provided. The author also addresses the problem of Liouvillian integration of factored quadratic systems that cannot be placed in the form above.