  We derive a new parametric class of exact cosmological solutions to
Brans-Dicke theory of gravity with a self-interacting scalar field and a
barotropic perfect fluid of ordinary matter, by assuming a linear relationship
between the Hubble expansion parameter and the time derivative of the scalar
field. As a consequence only a class of exponential potentials and their
combinations can be treated. The relevance of the solutions found for the
description of the cosmic evolution are discussed in detail. We focus our
discussion mainly on the possibility to have superquintessence behavior.
