In carpooling systems, a set of drivers owning a private car can accept a small detour to pick-up and drop-off other riders. However, carpooling is widely used for long-distance trips, where rider-driver matching can be done days ahead. Making carpooling a viable option for daily commute is more challenging, as trips are shorter and, proportionally, the detours tolerated by drivers are more tight. As a consequence, finding riders and drivers sharing close-enough origins, destinations and departure time is less likely, which limits potential matching. In this paper we propose an Integrated System, where carpooling matching is synchronized with Public Transit (PT) schedules, so as to serve as a feeder service to PT in the first mile. Driver detours are proposed towards PT selected stations, which are used as consolidation points, thus increasing matching probability. We present a computationally efficient method to represent PT schedules and drivers trajectory in a single General Transit Feed Specification database, which allows to compute multimodal rider journeys using any off the shelf planners. We showcase our approach in the metropolitan area of Portland, Oregon, considering 8k randomly generated trips. We show the benefits of our Integrated System. We find that 10% more riders find a feasible matching with respect to the status quo, where carpooling and PT are operated separately. We release our code as open source.