Goodwill and Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Mar. 31, 2024	Sep. 30, 2023
Goodwill (not amortized)	$ 3,446,373	$ 3,358,511
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,828,703	1,795,627
Intangible assets with finite lives (amortized), Accumulated Amortization	897,232	854,378
Intangible assets with finite lives (amortized), Net Book Value	931,471	941,249
Intangible Assets, Net (Including Goodwill)	4,377,844	4,299,760
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	627,704	615,915
Intangible assets with finite lives (amortized), Accumulated Amortization	415,352	395,109
Intangible assets with finite lives (amortized), Net Book Value	212,352	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	1,136,456	1,116,117
Intangible assets with finite lives (amortized), Accumulated Amortization	434,530	413,125
Intangible assets with finite lives (amortized), Net Book Value	701,926	702,992
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	37,770	36,851
Intangible assets with finite lives (amortized), Accumulated Amortization	20,577	19,400
Intangible assets with finite lives (amortized), Net Book Value	17,193	17,451
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	3,896	3,867
Intangible assets with finite lives (amortized), Accumulated Amortization	3,896	3,867
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0