Basement floor drain clogged with sediment/debris

I have a floor drain in my basement into which the water softener drains when it runs its backwash cycle. The drain has become completely clogged; sticking any kind of tool into it (screwdriver, snake, etc) meets resistance immediately, but it's like sticking a tool into sand. So I believe the drain is full of sediment or debris, possibly including the concrete basement wall, which is flaking to some extent nearby. I tried to get a plumber out but it's a small job and therefore isn't of much interest. Trying to figure out how to remove the debris myself. As I said, a hand snake doesn't work. What tool could I rent from home depot to clear the drain? And should I be focusing on removing the clog by destroying it or by suctioning it out somehow? Thanks for any suggestions.