We study a model of scalar field with a general non-minimal kinetic coupling to itself and to the curvature. The cosmological dynamics of this model and the issue of accelerated expansion are analyzed. Solutions giving rise to power-law expansion have been found. By constraining the potential of the model, we obtained a variety of solutions corresponding to phenomenologically acceptable models of dark energy, like the ΛCDM model, variable cosmological constant model VCC, FRW with two-dimensional topological defects and FRW with phantom dark energy, showing that this model with the appropriate potential in each case can describe a rich variety of dynamical scenarios.