  In the present work, we study slow-roll inflation in scalar-tensor gravity
theories in the presence of both the non-minimal coupling between the scalar
field and curvature, and the Galileon self-interaction of the scalar field.
Furthermore, we give predictions for the duration of reheating as well as for
the reheating temperature after inflation. After working out the expressions
for the power spectra of scalar and tensor perturbations in the case of a
general non-minimal coupling function that depends solely on the scalar field
and a general scalar potential, we focus on the special cases of the power-law
coupling function and chaotic quadratic inflation. Thus, under the slow-roll
approximation we confront the predictions of the model with the current PLANCK
constraints on the spectral index $n_s$ and the tensor-to-scalar ratio $r$
using the $n_{s}-r$ plane. We found that the combination of the non-minimal
coupling and Galileon self-interaction effects allows us to obtain better
results for $r$ than in the case in which each effect is considered separately.
Particularly, we obtained that the predictions of the model are in agreement
with the current observational bounds on $n_{s}$ and $r$ within the $95 \%$ C.L
region and also slightly inside the $68 \%$ C.L region. Also, we investigate
the oscillatory regime after the end of inflation by solving the full
background equations, and then we determine the upper bound for the Galileon
and non-minimal coupling parameters under the condition that the scalar field
oscillates coherently during reheating. Finally, after approximating reheating
by a constant equation of state, we derive the relations between the reheating
duration, the temperature at the end of reheating, its equation of state, and
the number of $e$-folds of inflation and then we relate them all to the
inflationary observables.
