Goodwill and Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Mar. 31, 2022	Sep. 30, 2021
Goodwill (not amortized)	$ 2,186,156	$ 2,191,887
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,111,901	1,112,070
Intangible assets with finite lives (amortized), Accumulated Amortization	756,541	733,103
Intangible assets with finite lives (amortized), Net Book Value	355,360	378,967
Intangible Assets, Net (Including Goodwill)	2,541,516	2,570,854
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	487,399	483,771
Intangible assets with finite lives (amortized), Accumulated Amortization	348,736	338,542
Intangible assets with finite lives (amortized), Net Book Value	138,663	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	570,920	574,516
Intangible assets with finite lives (amortized), Accumulated Amortization	363,623	350,648
Intangible assets with finite lives (amortized), Net Book Value	207,297	223,868
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	26,772	26,906
Intangible assets with finite lives (amortized), Accumulated Amortization	17,372	17,036
Intangible assets with finite lives (amortized), Net Book Value	9,400	9,870
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	3,933	4,000
Intangible assets with finite lives (amortized), Accumulated Amortization	3,933	4,000
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0