This exists. Its a program called Maple and really sophisticated. Industry standard. Every physicist/mathematician has a license. It can perform all the standard operations on a lot of math objects that you have probably never even heard about. No offense here. Math gets a lot stranger once you leave calculus behind.
There is wolframalpha, the computational knowledge engine. Provides maple core functionality and behaves more like a search engine.
If you do this for your own learning experience, that's awesome. As a business, you would have to be really good at a lot of very deep math to compete on that level.
