A power snake with a 1/4" cable is the preferred choice for clearing a bathtub drain. You can rent a quality top snake or mini power snake at a tool rental place. Often the rental fee will be considerably less than what it would cost to call a plumber.
If this is your first time using a power snake, be sure to ask for an explanation on how to use it. Each model can be a little different and you will want to be familiar with the proper use and safety features of your particular model. Taking a few minutes to learn how to use it can make the job considerably easier.