Goodwill and Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Mar. 31, 2021	Sep. 30, 2020
Goodwill (not amortized)	$ 2,193,632	$ 1,625,786
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,113,881	911,809
Intangible assets with finite lives (amortized), Accumulated Amortization	703,571	674,239
Intangible assets with finite lives (amortized), Net Book Value	410,310	237,570
Intangible Assets, Net (Including Goodwill)	2,603,942	1,863,356
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	484,190	443,275
Intangible assets with finite lives (amortized), Accumulated Amortization	322,869	309,124
Intangible assets with finite lives (amortized), Net Book Value	161,321	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	575,836	418,953
Intangible assets with finite lives (amortized), Accumulated Amortization	337,194	322,092
Intangible assets with finite lives (amortized), Net Book Value	238,642	96,861
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	26,959	22,687
Intangible assets with finite lives (amortized), Accumulated Amortization	16,612	16,129
Intangible assets with finite lives (amortized), Net Book Value	10,347	6,558
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	4,019	4,017
Intangible assets with finite lives (amortized), Accumulated Amortization	4,019	4,017
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0