  In this paper we consider the question of bounding the degree of an divisor
$D$ invariant by a $\F$ holomorphic foliation, without rational first integral,
on smooth algebraic variety $X$ in terms of degree of $\F$ and some invariants
of $D$ and $X$. Particularly, if $\F$ is a foliation of degree $d$ on
$\mathbb{P}_{\mathbb{C}}^2$, whose the number of invariants curves is greater
that ${k+2\choose k}$, we show that there exist a number $\mathcal{M}(d,k)$
such that if $k>\mathcal{M}(d,k),$ then $\F$ admits a rational first integral
of degree $\leq k$. Moreover, there exist a number $\mathscr{G}(d,k)$, such
that if $\F$ has an algebraic solution of degree $k$ and genus smaller than
$\mathscr{G}(d,k)$, then it has a rational first integral of degree $\leq k$.
