Goodwill and Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Jun. 30, 2022	Sep. 30, 2021
Goodwill (not amortized)	$ 2,382,680	$ 2,191,887
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,162,438	1,112,070
Intangible assets with finite lives (amortized), Accumulated Amortization	763,804	733,103
Intangible assets with finite lives (amortized), Net Book Value	398,634	378,967
Intangible Assets, Net (Including Goodwill)	2,781,314	2,570,854
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	503,579	483,771
Intangible assets with finite lives (amortized), Accumulated Amortization	352,587	338,542
Intangible assets with finite lives (amortized), Net Book Value	150,992	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	604,261	574,516
Intangible assets with finite lives (amortized), Accumulated Amortization	367,082	350,648
Intangible assets with finite lives (amortized), Net Book Value	237,179	223,868
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	27,865	26,906
Intangible assets with finite lives (amortized), Accumulated Amortization	17,402	17,036
Intangible assets with finite lives (amortized), Net Book Value	10,463	9,870
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	3,856	4,000
Intangible assets with finite lives (amortized), Accumulated Amortization	3,856	4,000
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0