  We present the implications for cosmic inflation of the Planck measurements
of the cosmic microwave background (CMB) anisotropies in both temperature and
polarization based on the full Planck survey. The Planck full mission
temperature data and a first release of polarization data on large angular
scales measure the spectral index of curvature perturbations to be
$n_\mathrm{s} = 0.968 \pm 0.006$ and tightly constrain its scale dependence to
$d n_s/d \ln k =-0.003 \pm 0.007$ when combined with the Planck lensing
likelihood. When the high-$\ell$ polarization data is included, the results are
consistent and uncertainties are reduced. The upper bound on the
tensor-to-scalar ratio is $r_{0.002} < 0.11$ (95% CL), consistent with the
B-mode polarization constraint $r< 0.12$ (95% CL) obtained from a joint
BICEP2/Keck Array and Planck analysis. These results imply that $V(\phi)
\propto \phi^2$ and natural inflation are now disfavoured compared to models
predicting a smaller tensor-to-scalar ratio, such as $R^2$ inflation. Three
independent methods reconstructing the primordial power spectrum are
investigated. The Planck data are consistent with adiabatic primordial
perturbations. We investigate inflationary models producing an anisotropic
modulation of the primordial curvature power spectrum as well as generalized
models of inflation not governed by a scalar field with a canonical kinetic
term. The 2015 results are consistent with the 2013 analysis based on the
nominal mission data.
