Collision detection is one of the most time-consuming operations during motion planning. Thus, there is an increasing interest in exploring machine learning techniques to speed up collision detection and sampling-based motion planning. A recent line of research focuses on utilizing neural signed distance functions of either the robot geometry or the swept volume of the robot motion. Building on this, we present a novel neural implicit swept volume model to continuously represent arbitrary motions parameterized by their start and goal configurations. This allows to quickly compute signed distances for any point in the task space to the robot motion. Further, we present an algorithm combining the speed of the deep learning-based signed distance computations with the strong accuracy guarantees of geometric collision checkers. We validate our approach in simulated and real-world robotic experiments, and demonstrate that it is able to speed up a commercial bin picking application.