Goodwill and Acquired Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Sep. 30, 2023	Sep. 30, 2022	Sep. 30, 2021
Goodwill (not amortized)	$ 3,358,511	$ 2,353,654	$ 2,191,887
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,795,627	1,152,018
Intangible assets with finite lives (amortized), Accumulated Amortization	854,378	769,300
Intangible assets with finite lives (amortized), Net Book Value	941,249	382,718
Intangible Assets, Net (Including Goodwill), Total	4,299,760	2,736,372
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	615,915	502,859
Intangible assets with finite lives (amortized), Accumulated Amortization	395,109	355,857
Intangible assets with finite lives (amortized), Net Book Value	220,806	147,002
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	1,116,117	594,970
Intangible assets with finite lives (amortized), Accumulated Amortization	413,125	369,390
Intangible assets with finite lives (amortized), Net Book Value	702,992	225,580
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	36,851	27,546
Intangible assets with finite lives (amortized), Accumulated Amortization	19,400	17,410
Intangible assets with finite lives (amortized), Net Book Value	17,451	10,136
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	3,867	3,766
Intangible assets with finite lives (amortized), Accumulated Amortization	3,867	3,766
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0