Goodwill and Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Dec. 31, 2021	Sep. 30, 2021
Goodwill (not amortized)	$ 2,189,880	$ 2,191,887
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,109,619	1,112,070
Intangible assets with finite lives (amortized), Accumulated Amortization	745,742	733,103
Intangible assets with finite lives (amortized), Net Book Value	363,877	378,967
Intangible Assets, Net (Including Goodwill)	2,553,757	2,570,854
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	482,824	483,771
Intangible assets with finite lives (amortized), Accumulated Amortization	344,152	338,542
Intangible assets with finite lives (amortized), Net Book Value	138,672	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	573,082	574,516
Intangible assets with finite lives (amortized), Accumulated Amortization	357,500	350,648
Intangible assets with finite lives (amortized), Net Book Value	215,582	223,868
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	26,864	26,906
Intangible assets with finite lives (amortized), Accumulated Amortization	17,241	17,036
Intangible assets with finite lives (amortized), Net Book Value	9,623	9,870
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	3,972	4,000
Intangible assets with finite lives (amortized), Accumulated Amortization	3,972	4,000
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0