  We apply analytic conformal bootstrap ideas in Mellin space to conformal
field theories with $O(N)$ symmetry and cubic anisotropy. We write down the
conditions arising from the consistency between the operator product expansion
and crossing symmetry in Mellin space. We solve the constraint equations to
compute the anomalous dimension and the OPE coefficients of all operators
quadratic in the fields in the epsilon expansion. We reproduce known results
and derive new results up to $O(\epsilon^3)$. For the $O(N)$ case, we also
study the large $N$ limit in general dimensions and reproduce known results at
the leading order in $1/N$.
