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We solve the Poincar\'e problem for plane foliations with only one dicriticaldivisor. Moreover, in this case, we give an algorithm that decides whether afoliation has a rational first integral and computes it in the affirmativecase. We also provide an algorithm to compute a rational first integral ofprefixed genus $g\neq 1$ of any type of plane foliation $\cf$. When the numberof dicritical divisors dic$(\cf)$ is larger than two, this algorithm depends onsuitable families of invariant curves. When dic$(\cf) = 2$, it proves that thedegree of the rational first integral can be bounded only in terms of $g$, thedegree of $\cf$ and the local analytic type of the dicritical singularities of$\cf$.

The Poincar\'e Problem, algebraic integrability and dicritical divisors