  In this paper, we investigate the Noether symmetries of a generalized
scalar-tensor, Brans-Dicke type cosmological model, in which we consider
explicit scalar field dependent couplings to the Ricci scalar, and to the
scalar field kinetic energy, respectively. We also include the scalar field
self-interaction potential into the gravitational action. From the condition of
the vanishing of the Lie derivative of the gravitational cosmological
Lagrangian with respect to a given vector field we obtain three cosmological
solutions describing the time evolution of a spatially flat
Friedman-Robertson-Walker Universe filled with a scalar field. The cosmological
properties of the solutions are investigated in detail, and it is shown that
they can describe a large variety of cosmological evolutions, including models
that experience a smooth transition from a decelerating to an accelerating
phase.
