One-dimensional anisotropic Heisenberg ferromagnetic spin chain can be described by the fifthorder nonlinear Schrödinger equation, which is investigated in this paper. Through the Darboux transformation, we obtain the Akhmediev breathers (ABs), Kuznetsov-Ma (KM) solitons and rogue-wave solutions. Effects of the coefficients of the fourth-order dispersion, γ , and of the fifth-order dispersion, δ, on the properties of ABs, KM solitons and rogue waves are discussed: (1) With γ increasing, the AB exhibits stronger localization in time; (2) The propagation directions of an AB and a KM soliton change with the presence of δ; and (3) Enhancement of γ makes the existence time of the rogue waves shorter, while enhancement of δ increases the existence time of the rogue waves.