In this paper, a variable coefficient high-order Schrödinger equation is studied to describe the propagation of ultrashort optical pulses in dispersion reducing fibers. It has third-order dispersion, self steepening and stimulated Raman scattering effects. Using the Bell polynomial method, the integrability of the variable coefficient higher-order Schrödinger equation is discussed. And the bilinear Bäcklund transformation, Lax pair, and infinite conservation laws of the equation are derived. It can be obtained that the variable coefficient higher-order Schrödinger equation is integrable in the sense of Lax pair. The soliton solutions are constructed from the obtained bilinear Bäcklund transformation and the bilinear form. The obtained results are analyzed in the initial–boundary value problem. It is worth mentioning that this paper, based on the angle of increasing the order, obtained the bilinear Bäcklund transformation of the third order variable coefficient Schrödinger equation by using the Bell polynomial method. On this basis, relevant properties such as Lax pair and infinite conservation laws are also obtained, and analyzed the application in the initial–boundary value problem. This has not been studied in existing literatures.