Algebraic integrability of foliations of the plane

We give an algorithm to decide whether an algebraic plane foliation F has a rational first integral and to compute it in the affirmative case. The algorithm runs whenever we assume the polyhedrality of the cone of curves of the surface obtained after blowing-up the set BF of infinitely near points needed to get the dicritical exceptional divisors of a minimal resolution of the singularities of F. This condition can be detected in several ways, one of them from the proximity relations in BF and, as a particular case, it holds when the cardinality of BF is less than 9.