Goodwill and Acquired Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Sep. 30, 2022	Sep. 30, 2021	Sep. 30, 2020
Goodwill (not amortized)	$ 2,353,654	$ 2,191,887	$ 1,625,786
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,152,018	1,112,070
Intangible assets with finite lives (amortized), Accumulated Amortization	769,300	733,103
Intangible assets with finite lives (amortized), Net Book Value	382,718	378,967
Intangible Assets, Net (Including Goodwill)	2,736,372	2,570,854
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	502,859	483,771
Intangible assets with finite lives (amortized), Accumulated Amortization	355,857	338,542
Intangible assets with finite lives (amortized), Net Book Value	147,002	145,229
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	594,970	574,516
Intangible assets with finite lives (amortized), Accumulated Amortization	369,390	350,648
Intangible assets with finite lives (amortized), Net Book Value	225,580	223,868
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	27,546	26,906
Intangible assets with finite lives (amortized), Accumulated Amortization	17,410	17,036
Intangible assets with finite lives (amortized), Net Book Value	10,136	9,870
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	3,766	4,000
Intangible assets with finite lives (amortized), Accumulated Amortization	3,766	4,000
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0