  In this article we show how to generalize to the Darbouxian, Liouvillian and
Riccati case the extactic curve introduced by J. Pereira. With this approach,
we get new algorithms for computing, if it exists, a rational, Darbouxian,
Liouvillian or Riccati first integral with bounded degree of a polynomial
planar vector field. We give probabilistic and deterministic algorithms. The
arithmetic complexity of our probabilistic algorithm is in
$\tilde{\mathcal{O}}(N^{\omega+1})$, where $N$ is the bound on the degree of a
representation of the first integral and $\omega \in [2;3]$ is the exponent of
linear algebra. This result improves previous algorithms. Our algorithms have
been implemented in Maple and are available on authors' websites. In the last
section, we give some examples showing the efficiency of these algorithms.
