Goodwill and Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Dec. 31, 2023	Sep. 30, 2023
Goodwill (not amortized)	$ 3,458,463	$ 3,358,511
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,834,464	1,795,627
Intangible assets with finite lives (amortized), Accumulated Amortization	881,094	854,378
Intangible assets with finite lives (amortized), Net Book Value	953,370	941,249
Intangible Assets, Net (Including Goodwill)	4,411,833	4,299,760
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	629,554	615,915
Intangible assets with finite lives (amortized), Accumulated Amortization	406,952	395,109
Intangible assets with finite lives (amortized), Net Book Value	222,602	220,806
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	1,140,213	1,116,117
Intangible assets with finite lives (amortized), Accumulated Amortization	427,224	413,125
Intangible assets with finite lives (amortized), Net Book Value	712,989	702,992
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	37,892	36,851
Intangible assets with finite lives (amortized), Accumulated Amortization	20,113	19,400
Intangible assets with finite lives (amortized), Net Book Value	17,779	17,451
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	3,928	3,867
Intangible assets with finite lives (amortized), Accumulated Amortization	3,928	3,867
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0