Income Taxes - Additional Information (Details) - USD ($)	12 Months Ended
	Sep. 30, 2024	Sep. 30, 2023	Sep. 30, 2022	Sep. 30, 2021
Income Tax Disclosure [Line Items]
Income taxes payable	$ 75,300,000	$ 30,400,000
Accrued income taxes	40,000,000	14,900,000
Prepaid income taxes	14,000,000	22,700,000
Income tax payments	68,600,000	65,900,000	$ 55,000,000
Foreign tax credits	2,028,000	3,750,000
Gross deferred amortization carryforward	66,100,000
Valuation allowance	21,755,000	21,695,000	22,283,000	$ 52,085,000
Interest Expense	119,653,000	129,417,000	54,268,000
Penalty expense	0	0	0
Interest expense related to income tax accruals	3,100,000	1,400,000
Accrued tax penalties	0	0	0
Income tax provision upon recognition of unrecognized tax benefit	65,000,000
Unrecognized tax benefits, increase in valuation allowance upon recognition	6,200,000
Potential decrease in unrecognized tax benefits	27,000,000
Stock-based compensation	223,461,000	206,459,000	174,863,000
Tax benefit recognition related to stock based compensation	27,500,000	33,400,000	27,100,000
Provision (benefit) for income taxes	92,629,000	87,026,000	84,017,000
Income Tax Expense
Income Tax Disclosure [Line Items]
Interest Expense	3,300,000	500,000	200,000
IRS Procedural Guidance Requiring Consent Domestic
Income Tax Disclosure [Line Items]
Provision (benefit) for income taxes	4,400,000
Domestic Country
Income Tax Disclosure [Line Items]
Valuation allowance	$ 17,400,000
Domestic Country | Minimum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2026
Domestic Country | Maximum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2044
Domestic Country | Expire in 2025 to 2034
Income Tax Disclosure [Line Items]
Net operating loss carryforwards	$ 800,000
Domestic Country | Expiration Beginning in 2026 and Ending in 2044
Income Tax Disclosure [Line Items]
Credit carryforwards	4,600,000
State - Massachusetts
Income Tax Disclosure [Line Items]
Valuation allowance	$ 17,400,000
State - Massachusetts | Minimum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2025
State - Massachusetts | Maximum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2039
State - Massachusetts | Expiration Beginning in 2027
Income Tax Disclosure [Line Items]
Net operating loss carryforwards	$ 6,900,000
State - Massachusetts | Research Tax Credit Carryforward
Income Tax Disclosure [Line Items]
Credit carryforwards	29,200,000
Foreign Country
Income Tax Disclosure [Line Items]
Net operating loss carryforwards	6,900,000
Credit carryforwards	2,000,000
Foreign tax credits	2,200,000
Valuation allowance	4,400,000
Provision (benefit) for income taxes	4,600,000
Tax credits	$ 14,400,000
Foreign Country | Minimum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2032
Non-U.S. tax credit carryforwards expiration	2031
Foreign Country | Maximum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2034
Non-U.S. tax credit carryforwards expiration	2042
Windfall Tax Benefit
Income Tax Disclosure [Line Items]
Provision (benefit) for income taxes	$ (10,200,000)	(7,800,000)	$ (5,200,000)
Transfer Pricing Foreign Jurisdiction
Income Tax Disclosure [Line Items]
Uncertain tax		21,800,000
Other Current Liabilities
Income Tax Disclosure [Line Items]
Accrued income taxes	6,200,000	4,800,000
Other Liabilities [Member]
Income Tax Disclosure [Line Items]
Accrued income taxes	$ 29,100,000	$ 10,700,000