We have developed a new model for predicting the time-dependent radiation drive asymmetry in laser-heated hohlraums. The model consists of integrated Hydra capsule-hohlraum calculations coupled to a separate model for calculating the crossbeam energy transfer between the inner and outer cones of the National Ignition Facility (NIF) indirect drive configuration. The time- dependent crossbeam transfer model parameters were adjusted in order to best match the P2 component of the shape of the inflight shell inferred from backlit radiographs of the capsule taken when the shell was at a radius of 150-250 μm. The adjusted model correctly predicts the observed inflight P2 and P4 components of the shape of the inflight shell, and also the P2 component of the shape of the hotspot inferred from x-ray self-emission images at the time of peak emission. It also correctly captures the scaling of the inflight P4 as the hohlraum length is varied. We then applied the newly benchmarked model to quantify the improved symmetry of the N130331 layered deuterium- tritium (DT) experiment in a re-optimized longer hohlraum.