It depends on a few different things, including the content, the style, and when it was made (i.e. did the technology exist to do it one way vs. another?)
Here's a GREAT example of multiple anime styles used in a single show -- we go from "high quality" level animation for the transformations, to the more "western", flash-friendly style the show is normally in, to a rather garish live-action explosion at the end. These all require different approaches.
If you pay attention, you can also spot where the characters aren't moving much in certain points in the video (like where Panty & Stocking are standing still under the "disco" lighting), so they can get away with only drawing the frame once even though the scene goes for much longer.
When "tweening" between keyframes (to produce moving animation like your soccer ball example), yes, those have to be hand-drawn -- or at least hand-checked, even if a computer is used to extrapolate between keyframes! Using certain animation software like Flash, Toon Boom, or 3D modeling software like Blender may take out some of the work, but none of that software can compensate for a bad animator, or replace an artist at the helm.
There's plenty of shortcuts animators use, but the computer never draws the cartoon for them -- they still have to do that part themselves!