  We propose a dark energy model where a scalar field $\phi$ has nonlinear
self-interactions in the presence of a dilatonic coupling with the Ricci
scalar. This belongs to a sub-class of theories beyond Horndeski, which
accommodates covariant Galileons and Brans-Dicke theories as specific cases. We
derive conditions under which the scalar sound speed squared $c_{\rm s}^2$ is
positive from the radiation era to today. Since $c_{\rm s}^2$ remains to be
smaller than the order of 1, the deviation from Horndeski theories does not
cause heavy oscillations of gauge-invariant gravitational potentials. In this
case, the evolution of matter perturbations at low redshifts is similar to that
in the coupled dark energy scenario with an enhanced gravitational interaction.
On the spherically symmetric background with a matter source, the existence of
field self-interactions suppresses the propagation of fifth force inside a
Vainshtein radius. We estimate an allowed parameter space in which the model
can be compatible with solar system constraints while satisfying conditions for
the cosmological viability of background and perturbations.
