Autonomous off-road navigation requires robots to estimate terrain traversability from onboard sensors and plan motion accordingly. Conventional approaches typically rely on sampling-based planners such as MPPI to generate short-term control actions that aim to minimize traversal time and risk measures derived from the traversability estimates. These planners can react quickly but optimize only over a short look-ahead window, limiting their ability to reason about the full path geometry, which is important for navigating in challenging off-road environments. Moreover, they lack the ability to adjust speed based on the terrain-induced vibrations, which is important for smooth navigation on challenging terrains. In this paper, we introduce TRAIL (Traversability with an Implicit Learned Representation), an off-road navigation framework that leverages an implicit neural representation to model terrain properties as a continuous field that can be queried at arbitrary locations. This representation yields spatial gradients that enable integration with a novel gradient-based trajectory optimization method that adapts the path geometry and speed profile based on terrain traversability.