  Here we present an efficient method to compute Darboux polynomials for
polynomial vector fields in the plane. This approach is restricetd to
polynomial vector fields presenting a Liouvillian first integral (or,
equivalently, to rational first order differential equations (rational 1ODEs)
presenting a Liouvillian general solution). The key to obtaining this method
was to separate the procedure of solving the (nonlinear) algebraic systems
resulting from the equation that translates the condition of existence of a
Darboux polynomial into feasible steos (procedures that requires less memory
consumption). We also present a brief performance analysis of the algorithms
developed.
