Goodwill and Acquired Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Sep. 30, 2021	Sep. 30, 2020	Sep. 30, 2019
Goodwill (not amortized)	$ 2,191,887	$ 1,625,786	$ 1,238,179
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,112,070	911,809
Intangible assets with finite lives (amortized), Accumulated Amortization	733,103	674,239
Intangible assets with finite lives (amortized), Net Book Value	378,967	237,570
Intangible assets	2,570,854	1,863,356
Intangible Assets, Net (Including Goodwill)	2,570,854	1,863,356
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	483,771	443,275
Intangible assets with finite lives (amortized), Accumulated Amortization	338,542	309,124
Intangible assets with finite lives (amortized), Net Book Value	145,229	134,151
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	574,516	418,953
Intangible assets with finite lives (amortized), Accumulated Amortization	350,648	322,092
Intangible assets with finite lives (amortized), Net Book Value	223,868	96,861
Trademarks
Intangible assets with finite lives (amortized), Gross Carrying Amount	26,906	22,687
Intangible assets with finite lives (amortized), Accumulated Amortization	17,036	16,129
Intangible assets with finite lives (amortized), Net Book Value	9,870	6,558
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	4,000	4,017
Intangible assets with finite lives (amortized), Accumulated Amortization	4,000	4,017
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0