  We construct certain higher order smooth positon and breather positon
solutions of an extended nonlinear Schr\"odinger equation with the cubic and
quartic nonlinearity. We utilize the generalized Darboux transformation method
to construct the aforementioned solutions. The three well-known equations,
namely nonlinear Schr\"odinger equation, Hirota equation, and generalized
nonlinear Schr\"odinger equation, are sub-cases of the considered extended
nonlinear Schr\"odinger equation. The solutions which we construct are more
general. We analyze how the positon and breather positon solutions of the
constituent equations get modified by the higher order nonlinear and dispersion
terms. Our results show that the width and direction of the smooth positon and
breather-positon solutions are highly sensitive to higher-order effects.
Further, we carryout an asymptotic analysis to predict the behaviour of
positons. We observe that during collision positons exhibit a time-dependent
phase shift. We also present the exact expression of time-dependent phase shift
of positons. Finally, we show that this time-dependent phase shift is directly
proportional to the higher order nonlinear and dispersion parameters.
