If water is backing up into your bathtub, the culprit is most likely a blockage - not a leak.
Have you tried conventional declogging methods, like a draining solution or a snake that might free up the blockage? If not, try these easy remedies. If they're unsuccessful, and if a plumber continues to come up short, consider a drain camera. They are typically used by sewer cleaning companies as opposed to plumbers, and are small cameras snaked up into the drain from the street side that can reveal the problem.
My guess is that you're dealing with one of two things: A garden variety physical obstruction, such as roots or buildup, or a crack in the drain pipe that's allowed outside blockage to creep in. Broken pipes have rough edges, so they're very fertile for growth including sewage and other organic buidup from both inside and outside of the pipes. If it turns out the pipe IS cracked, and if you'd prefer not to dig up the surface above it, look into inserting new pipe inside the old one. Like a fiberglass sock of sorts, the new pipe is inserted into the existing pipe and expands with hot water, basically creating a new pipe liner.