  We investigate the field equations of the conformally invariant models of
gravity with curvature-matter coupling, constructed in Weyl geometry, by using
the Palatini formalism. We consider the case in which the Lagrangian is given
by the sum of the square of the Weyl scalar, of the strength of the field
associated to the Weyl vector, and a conformally invariant geometry-matter
coupling term, constructed from the matter Lagrangian and the Weyl scalar.
After substituting the Weyl scalar in terms of its Riemannian counterpart, the
quadratic action is defined in Riemann geometry, and involves a nonminimal
coupling between the Ricci scalar and the matter Lagrangian. For the sake of
generality, a more general Lagrangian, in which the Weyl vector is nonminmally
coupled with an arbitrary function of the Ricci scalar, is also considered. By
varying the action independently with respect to the metric and the connection,
the independent connection can be expressed as the Levi-Civita connection of an
auxiliary, Ricci scalar and Weyl vector dependent metric, which is related to
the physical metric by means of a conformal transformation. The field equations
are obtained in both the metric and the Palatini formulations. The cosmological
implications of the Palatini field equations are investigated for three
distinct models corresponding to different forms of the coupling functions. A
comparison with the standard $\Lambda$CDM model is also performed, and we find
that the Palatini type cosmological models can give an acceptable description
of the observations.
