  We analytically study rogue-wave (RW) solutions and rational solitons of an
integrable fifth-order nonlinear Schr\"odinger (FONLS) equation with three free
parameters. It includes, as particular cases, the usual NLS, Hirota, and
Lakshmanan-Porsezian-Daniel (LPD) equations. We present continuous-wave (CW)
solutions and conditions for their modulation instability in the framework of
this model. Applying the Darboux transformation to the CW input, novel first-
and second-order RW solutions of the FONLS equation are analytically found. In
particular, trajectories of motion of peaks and depressions of profiles of the
first- and second-order RWs are produced by means of analytical and numerical
methods. The solutions also include newly found rational and W-shaped one- and
two-soliton modes. The results predict the corresponding dynamical phenomena in
extended models of nonlinear fiber optics and other physically relevant
integrable systems.
