The first integrals of the fifth order nonlinear differential equations

The object of study is a nonlinear fifth-order differential equation with a rational right-hand side. The goal of the study is to find the conditions for the existence of the first integrals of the equation and study the analytical properties of the obtained first integrals. In main part of the nonlinear differential fifth-order equation obtained the equations for residues and the resonance equation. Depending on the roots of the resonance equation, the necessary conditions of existence of the first integrals of nonlinear differential equation of the fifth order with a rational right-hand side is found. For these differential equation corresponding first integrals considered in this paper special cases are given. The obtained results can be used in the analytical theory of the ordinary differential equations.