Goodwill and Acquired Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Sep. 30, 2020	Sep. 30, 2019	Sep. 30, 2018
Goodwill (not amortized)	$ 1,625,786	$ 1,238,179	$ 1,182,457
Intangible assets with finite lives (amortized), Gross Carrying Amount	911,809	778,968
Intangible assets with finite lives (amortized), Accumulated Amortization	674,239	609,019
Intangible assets with finite lives (amortized), Net Book Value	237,570	169,949
Intangible Assets, Net (Including Goodwill)	1,863,356	1,408,128
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	443,275	377,359
Intangible assets with finite lives (amortized), Accumulated Amortization	309,124	278,144
Intangible assets with finite lives (amortized), Net Book Value	134,151	99,215
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	418,953	355,931
Intangible assets with finite lives (amortized), Accumulated Amortization	322,092	288,828
Intangible assets with finite lives (amortized), Net Book Value	96,861	67,103
Trademarks
Intangible assets with finite lives (amortized), Gross Carrying Amount	22,687	18,891
Intangible assets with finite lives (amortized), Accumulated Amortization	16,129	15,260
Intangible assets with finite lives (amortized), Net Book Value	6,558	3,631
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	4,017	3,910
Intangible assets with finite lives (amortized), Accumulated Amortization	4,017	3,910
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0