Control barrier functions (CBFs) are a standard way to keep a planner inside a safe set, but they assume control affine dynamics, a barrier function given in closed form, and a known nominal controller. Integral control barrier functions (ICBFs) remove the first assumption and absorb input constraints, yet they still assume a given nominal controller and a barrier function. In this work, we present an approach to learn all three from data, using neural ordinary differential equations (ODEs) for the dynamics, imitation for the nominal controller, and supervised regression for the barrier function. The learned dynamics, nominal controller, and the barrier function are used to build the ICBF filter constraint analytically. Furthermore, the barrier is trained on viability rather than geometry, meaning that a state counts as safe only if some emergency maneuver can still avoid the constraint under the actuator limits, a correction that geometric labels miss entirely and that grows with speed. Actuator limits then enter the online program as hard bounds while the learned barrier rows may be relaxed, which keeps the program solvable at every state and the actuator limits exact, and we bound how far the input it returns can drift from the one the true models would give. On agile racing with a 6-D dynamic bicycle, the filter removes all 12 unsafe episodes from 200 held out layouts at an unchanged success rate and above 100 Hz on one CPU core, holds the input constraint on every episode of a benchmark where a jointly trained baseline holds it on 53.38%, and keeps that advantage under large errors in the plant friction.