Traditional compressive X-ray tomosynthesis uses sequential illumination to interrogate the object, leading to long scanning time and image distortion due to the object variation. This paper proposes a single-snapshot compressive tomosynthesis imaging approach, where the object is simultaneously illuminated by multiple X-ray emitters equipped with coded apertures. Based on rank, intensity and sparsity prior models, a nonlinear image reconstruction framework is established. The coded aperture patterns are optimized based on uniform sensing criteria. Then, a modified split Bregman algorithm is developed to reconstruct the object from the set of nonlinear compressive measurements. It is shown that the proposed method can be used to reduce the inspection time and achieve robust reconstruction with respect to shape variation or motion of objects.