Goodwill and Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Dec. 31, 2022	Sep. 30, 2022
Goodwill (not amortized)	$ 2,388,589	$ 2,353,654
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,168,879	1,152,018
Intangible assets with finite lives (amortized), Accumulated Amortization	795,042	769,300
Intangible assets with finite lives (amortized), Net Book Value	373,837	382,718
Intangible Assets, Net (Including Goodwill)	2,762,426	2,736,372
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	508,840	502,859
Intangible assets with finite lives (amortized), Accumulated Amortization	366,149	355,857
Intangible assets with finite lives (amortized), Net Book Value	142,691	147,002
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	605,366	594,970
Intangible assets with finite lives (amortized), Accumulated Amortization	384,198	369,390
Intangible assets with finite lives (amortized), Net Book Value	221,168	225,580
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	27,911	27,546
Intangible assets with finite lives (amortized), Accumulated Amortization	17,933	17,410
Intangible assets with finite lives (amortized), Net Book Value	9,978	10,136
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	3,885	3,766
Intangible assets with finite lives (amortized), Accumulated Amortization	3,885	3,766
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0