Goodwill and Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Dec. 31, 2020	Sep. 30, 2020
Goodwill (not amortized)	$ 1,635,281	$ 1,625,786
Intangible assets with finite lives (amortized), Gross Carrying Amount	919,868	911,809
Intangible assets with finite lives (amortized), Accumulated Amortization	693,972	674,239
Intangible assets with finite lives (amortized), Net Book Value	225,896	237,570
Intangible Assets, Net (Including Goodwill)	1,861,177	1,863,356
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	446,336	443,275
Intangible assets with finite lives (amortized), Accumulated Amortization	317,753	309,124
Intangible assets with finite lives (amortized), Net Book Value	128,583	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	423,719	418,953
Intangible assets with finite lives (amortized), Accumulated Amortization	332,787	322,092
Intangible assets with finite lives (amortized), Net Book Value	90,932	96,861
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	22,854	22,687
Intangible assets with finite lives (amortized), Accumulated Amortization	16,473	16,129
Intangible assets with finite lives (amortized), Net Book Value	6,381	6,558
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	4,082	4,017
Intangible assets with finite lives (amortized), Accumulated Amortization	4,082	4,017
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0