Goodwill and Intangible Assets - Schedule of Goodwill and Acquired Intangible Assets (Details) - USD ($) $ in Thousands	Dec. 31, 2024	Sep. 30, 2024
Goodwill (not amortized)	$ 3,423,420	$ 3,461,891
Intangible assets with finite lives (amortized), Gross Carrying Amount	1,822,484	1,840,304
Intangible assets with finite lives (amortized), Accumulated Amortization	950,376	942,828
Intangible assets with finite lives (amortized), Net Book Value	872,108	897,476
Intangible Assets, Net (Including Goodwill)	4,295,528	4,359,367
Purchased Software
Intangible assets with finite lives (amortized), Gross Carrying Amount	628,414	634,439
Intangible assets with finite lives (amortized), Accumulated Amortization	440,693	436,471
Intangible assets with finite lives (amortized), Net Book Value	187,721	197,968
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	1,129,789	1,141,086
Intangible assets with finite lives (amortized), Accumulated Amortization	460,818	457,718
Intangible assets with finite lives (amortized), Net Book Value	668,971	683,368
Trademarks and Trade Names
Intangible assets with finite lives (amortized), Gross Carrying Amount	37,565	37,961
Intangible assets with finite lives (amortized), Accumulated Amortization	22,149	21,821
Intangible assets with finite lives (amortized), Net Book Value	15,416	16,140
Other
Intangible assets with finite lives (amortized), Gross Carrying Amount	3,839	3,941
Intangible assets with finite lives (amortized), Accumulated Amortization	3,839	3,941
Intangible assets with finite lives (amortized), Net Book Value	$ 0	$ 0