Goodwill and Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Jun. 30, 2024	Sep. 30, 2023
Goodwill (not amortized)	$ 3,442,245	$ 3,358,511
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,826,656	1,795,627
Intangible assets with finite lives (amortized), Accumulated Amortization	916,151	854,378
Intangible assets with finite lives (amortized), Net Book Value	910,505	941,249
Intangible Assets, Net (Including Goodwill)	4,352,750	4,299,760
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	627,099	615,915
Intangible assets with finite lives (amortized), Accumulated Amortization	424,643	395,109
Intangible assets with finite lives (amortized), Net Book Value	202,456	220,806
Capitalized Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	22,877	22,877
Intangible assets with finite lives (amortized), Accumulated Amortization	22,877	22,877
Intangible assets with finite lives (amortized), Net Book Value	0	0
Customer Lists and Relationships
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,135,067	1,116,117
Intangible assets with finite lives (amortized), Accumulated Amortization	443,644	413,125
Intangible assets with finite lives (amortized), Net Book Value	691,423	702,992
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	37,727	36,851
Intangible assets with finite lives (amortized), Accumulated Amortization	21,101	19,400
Intangible assets with finite lives (amortized), Net Book Value	16,626	17,451
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	3,886	3,867
Intangible assets with finite lives (amortized), Accumulated Amortization	3,886	3,867
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0