  We investigate the cosmological observational test of the extended
quintessence model, i.e. a scalar-tensor gravity model with a scalar field
potential serving as dark energy, by using the Planck 2018 cosmic microwave
background (CMB) data, together with the baryon acoustic oscillations (BAO) and
redshift-space distortion (RSD) data. As an example, we consider the model with
a Brans-Dicke kinetic term $\frac{\omega(\phi)}{\phi} \phi_{;\mu} \phi^{;\mu} $
and a quadratic scalar potential $V (\phi) = A + B (\phi - \phi_0) +
\frac{C}{2} (\phi - \phi_0)^2$, which reduces to general relativity (GR) in the
limit $\omega(\phi) \to \infty$, and the cosmological constant in the limit
$B=C=0$. In such a model the scalar field typically rolls down the potential
and oscillates around the minimum of $V (\phi)$. We find that the model
parameter estimate for the CMB+BAO+RSD data set is given by $\lg \alpha = -3.6
_{-0.54}^{+0.66}~ (68\%)$, corresponding to $ 3.8 \times 10^5 < \omega_0 < 9.5
\times 10^7~ (68\%)$, and $\lg C = 4.9 \pm 1.4~ (68\%) $. However, the GR
$\Lambda$CDM model can fit the data almost as good as this extended
quintessence model, and is favored by the Akaike information criterion (AIC).
The variation of the gravitational constant since the epoch of Recombination is
constrained to be $0.97 < G_{\rm rec}/G_0 < 1.03~ (1 \sigma)$. In light of
recent report that the CMB data favors a closed universe, we consider the case
with non-flat geometry in our fit, and find that the mean value of $\Omega_k$
shifts a little bit from $-0.049$ to $-0.036$, and the parameters in our model
are not degenerate with $\Omega_k$.
