Bacteria such as Escherichia coli move about in a series of runs and tumbles: while a run state (straight motion) entails all the flagellar motors spinning in counterclockwise (CCW) mode, a tumble is caused by a shift in the state of one or more motors to clockwise (CW) spinning mode. In the presence of an attractant gradient in the environment, runs in the favourable direction are extended, and this results in a net drift of the organism in the direction of the gradient. Existing theoretical predictions for the drift velocity are limited to exponentially distributed run durations. However, recent experimental observations strongly suggest that the CCW and CW intervals have gamma, rather than exponential distributions. We present a path-integral method which can be used to compute various quantities of interest for the run and tumble walk, with and without chemotaxis, for arbitrary distributions of run and tumble intervals, as power series expansions in the gradient. The effectiveness of the method is demonstrated by deriving a number of existing results for the mean-squared displacement (including motion with directional persistence and algebraically distributed run times) and also chemotactic drift (with exponentially distributed run intervals) in a systematic way, starting from a set of general formulae. New results for chemotactic drift velocity for gamma-distributed run and tumble intervals are then derived, in the limit of weak gradients. Finally, by making use of available experimental data, we make testable predictions for the dependence of the drift velocity on the clockwise bias of the flagellar motor. © 2019 IOP Publishing Ltd.