busy has 9 distinct synonyms spread across 4 of its 6 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. Its one recorded opposite is idle.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
actively or fully engaged or occupied
“busy with her work”
intrusive in a meddling or offensive manner
“an interfering old woman”
1 of the 6 recorded senses carries no synonym, antonym or related term, so it is not shown above. Every sense, with its definition and examples, is on the full entry for busy.
Synonyms, antonyms and the broader and narrower relations for busy come from Princeton University’s WordNet. Full sources and licences.