Interval censoring frequently occurs in many biomedical or health studies, and an interval containing exact failure time is observed from sequential monitoring. When exact failure time is also partly available, it is often referred as double and partly interval censoring, and it has been studied by several authors while they focused on transformation models. In this paper, we suggest the semiparametric accelerated failure time model under double and partly interval censoring. Gehan-type weighted estimating function is constructed by investigating comparable rank cases for each censoring indicator, and extension to the general class of estimating function can be simply derived. Furthermore, an efficient variance estimation procedure is considered to reduce computation time compared to existing resampling methods. Asymptotic behaviors of the method are established under mild conditions by using standard empirical processes. Simulation studies demonstrate the method is more efficient than existing Buckley-James methods in dominant cases. Two data examples are given to illustrate the practical usefulness of the method.