  In this paper, we construct a class of (n+1)-dimensional $(n\geq4)$ slowly
rotating black hole solutions in Brans-Dicke-Maxwell theory with a quadratic
potential. These solutions can represent black holes with inner and outer event
horizons, an extreme black hole and a naked singularity and they are neither
asymptotically flat nor (anti)-de Sitter. We compute the Euclidean action and
use it to obtain the conserved and thermodynamics quantities such as entropy,
which does not obey the area law. We also compute the angular momentum and the
gyromagnetic ratio for these type of black holes where the gyromagnetic ratio
is modified in Brans-Dicke theory compared to the Einstein theory.
