© 2016 American Physical Society. We consider the application of group invariant transformations in order to constrain a flat isotropic and homogeneous cosmological model, containing a Brans-Dicke scalar field and a perfect fluid with a constant equation of state parameter w, where the latter is not interacting with the scalar field in the gravitational action integral. The requirement that the Wheeler-DeWitt equation be invariant under one-parameter point transformations provides us with two families of power-law potentials for the Brans-Dicke field, in which the powers are functions of the Brans-Dicke parameter ωBD and the parameter w. The existence of the Lie symmetry in the Wheeler-DeWitt equation is equivalent to the existence of a conserved quantity in field equations and with oscillatory terms in the wave function of the Universe. This enables us to solve the field equations. For a specific value of the conserved quantity, we find a closed-form solution for the Hubble factor, which is equivalent to a cosmological model in general relativity containing two perfect fluids. This provides us with different models for specific values of the parameters ωBD, and w. Finally, the results hold for the specific case where the Brans-Dicke parameter ωBD is zero, that is, for the O'Hanlon massive dilaton theory and, consequently, for f(R) gravity in the metric formalism.