© 2025 the author(s).Using the normal form theory and the existence of an algebraic inverse integrating factor we c(haracterize the local analytic integrability of the systems whose quasi-homogeneous leading term is (a1 y3 + a2x3 y, b1x5 + b2x2 y2). More specifically we prove that the analytic integrable vector fields inside such family are orbitally equivalent to a semi-quasi-homogeneus system, that is, are not orbitally equivalent to its lowest-degree quasi-homogeneous term.