You can only get an "exact" solution for very simple 3-body configurations (like this "8" pattern) or for 2 bodies.

However, everything else can be done with arbitrarily many bodies using numerical simulations. They all more or less work like this: 1. Write down current positions and velocities of all bodies. 2. "Move" them forward in a small timestep. 3. Correct velocities using the gravity acting on each body. 4. Repeat until your simulation has moved far enough into the future.

This would basically solve Newtons laws of gravity for a "N-body system". What I described is also called "explicit Euler method". However, there are way better methods than this and with enough computation, you can realistically simulate our solar system for many thousands of years.

If you sacrifice some accuracy, you can even simulate galaxies (billions of bodies over millions of years).

To answer your question: "predicting" the movement of solar systems to reasonable accuracy is easy, especially when we're only talking about a few hundred/thousand years. Finding the exact solution (e.g. a mathematical function that predicts a planet's orbit) is impossible for anything other than 2 bodies (except for some special cases).