  We study the class of planar polynomial vector fields admitting Darboux first
integrals of the type $\prod_{i=1}^r f_i^{\alpha_i}$, where the $\alpha_i$'s
are positive real numbers and the $f_i$'s are polynomials defining curves with
only one place at infinity. We show that these vector fields have an extended
reduction procedure and give an algorithm which, from a part of the extended
reduction of the vector field, computes a Darboux first integral for generic
exponents.
