Income Taxes - Schedule of Deferred Tax Assets and Liabilities (Details) - USD ($) $ in Thousands	Sep. 30, 2024	Sep. 30, 2023	Sep. 30, 2022	Sep. 30, 2021
Deferred Tax Assets, Net [Abstract]
Net operating loss carryforwards	$ 14,141	$ 22,272
Foreign tax credits	2,028	3,750
Capitalized research and development	136,001	83,748
Pension benefits	7,629	5,327
Prepaid expenses	18,551	15,040
Deferred revenue	2,607	5,902
Stock-based compensation	22,231	19,684
Other reserves not currently deductible	34,422	19,604
Amortization of intangible assets	60,527	90,888
Research and development and other tax credits	25,706	64,618
Lease liabilities	46,460	50,102
Fixed assets	106,741	83,796
Capital loss carryforward	3,875	3,700
Other	3,528	2,279
Gross deferred tax assets	484,447	470,710
Valuation allowance	(21,755)	(21,695)	$ (22,283)	$ (52,085)
Total deferred tax assets	462,692	449,015
Deferred Tax Liabilities, Net [Abstract]
Acquired intangible assets not deductible	(257,731)	(263,178)
Lease assets	(34,160)	(37,332)
Pension prepayments	(3,283)	(1,808)
Deferred revenue	(1,243)	(17,400)
Depreciation	(4,683)	(5,779)
Deferred income	(11,636)	(8,656)
Prepaid commissions	(13,738)	(13,757)
Other	(9,030)	(7,294)
Total deferred tax liabilities	(335,504)	(355,204)
Net deferred tax assets	$ 127,188	$ 93,811