Cut down into the concrete slab around where the section of damp concrete that is indicative of a leak. Most basement slabs are between 4 and 6 inches thick, so cut down roughly 4 inches to start. Cut a 1-foot square hole. Place the blade of the saw against the concrete, pull the trigger and slowly sink the blade into the concrete. If your saw has a blade gauge, set the gauge before cutting to control the depth.
Use a jackhammer to break up the concrete within the square you cut with the concrete saw. Remove the pieces. Work your way down until you reach the broken section of pipe and clear all of the concrete away from the pipe as well as to either side of the broken section. Remove more concrete as necessary with the concrete saw and the jackhammer.
Remove the excess water from the area with a wet vacuum. Cut away the damaged section of PVC pipe with a hacksaw and remove it. Let the area dry for at least 24 hours to remove the moisture from the trench and the pipes, or use an old rag (or multiple rags) to completely soak up the moisture since the pipes need to be completely dry to stick together.
Dry fit the two couplers onto either section of the existing pipe. Cut a new section of PVC pipe down to size with the hacksaw to fit into the couplers. Remove the couplers and the new section. Prime the male/female sections of the couplers and pipes that are being attached together. Apply pipe cement to the same sections and connect everything together.
Mix up a batch of concrete mix in the bucket with your drill and paddle attachment. Stir it until you have a thick, creamy mixture. Pour this new concrete into the hole on top of the repaired pipe. Fill the hole completely and then use a flat metal trowel to smooth the concrete surface to match the surrounding concrete. Allow 72 hours of drying before you allow foot traffic.