Income Taxes - Additional Information (Details) - USD ($)	12 Months Ended
	Sep. 30, 2023	Sep. 30, 2022	Sep. 30, 2021	Sep. 30, 2020
Income Tax Disclosure [Line Items]
Income taxes payable	$ 30,400,000	$ 17,300,000
Accrued income taxes	14,900,000	5,100,000
Prepaid income taxes	22,700,000	25,800,000
Income tax payments	65,900,000	55,000,000	$ 56,000,000
Foreign tax credits	3,750,000	14,527,000
Gross deferred amortization carryforward	85,300,000
Valuation allowance	21,695,000	22,283,000	52,085,000	$ 205,423,000
Interest Expense	129,417,000	54,268,000	50,478,000
Penalty expense	0	0	2,000,000
Interest expense related to income tax accruals	1,400,000	900,000
Accrued tax penalties	0	0	0
Income tax provision upon recognition of unrecognized tax benefit	50,700,000
Unrecognized tax benefits, increase in valuation allowance upon recognition	5,700,000
Potential decrease in unrecognized tax benefits	26,000,000
Charge related to effects of Non US Jurisdiction Tax Matter			37,300,000
US Income taxes on repatriated foreign earnings			2,900,000
Stock-based compensation	206,459,000	174,863,000	177,289,000
Tax benefit recognition related to stock based compensation	33,400,000	27,100,000	39,900,000
Provision (benefit) for income taxes	87,026,000	84,017,000	(85,168,000)
Income Tax Expense
Income Tax Disclosure [Line Items]
Interest Expense	500,000	200,000	2,200,000
Domestic Country
Income Tax Disclosure [Line Items]
Net operating loss carryforwards	6,400,000
Valuation allowance	$ 17,400,000
Domestic Country | Minimum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2025
Domestic Country | Maximum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2043
Domestic Country | Expire in 2023 to 2036
Income Tax Disclosure [Line Items]
Net operating loss carryforwards	$ 1,200,000
Credit carryforwards	46,700,000
Domestic Country | Not Expire
Income Tax Disclosure [Line Items]
Net operating loss carryforwards	5,200,000
State - Massachusetts
Income Tax Disclosure [Line Items]
Valuation allowance	$ 17,400,000
State - Massachusetts | Minimum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2024
State - Massachusetts | Maximum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2038
State - Massachusetts | Research Tax Credit Carryforward
Income Tax Disclosure [Line Items]
Credit carryforwards	$ 27,300,000
Foreign Country
Income Tax Disclosure [Line Items]
Net operating loss carryforwards	6,800,000
Credit carryforwards	3,700,000
Foreign tax credits	2,300,000
Valuation allowance	$ 4,300,000
Foreign Country | Minimum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2030
Non-U.S. tax credit carryforwards expiration	2030
Foreign Country | Maximum
Income Tax Disclosure [Line Items]
Credit carryforwards expiration	2033
Non-U.S. tax credit carryforwards expiration	2041
REPUBLIC OF KOREA
Income Tax Disclosure [Line Items]
Charge related to effects of Non US Jurisdiction Tax Matter			34,400,000
Payment related to effects of Non US jurisdiction tax matter			20,000,000
Windfall Tax Benefit
Income Tax Disclosure [Line Items]
Provision (benefit) for income taxes	$ (7,800,000)	(5,200,000)	$ (9,900,000)
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	4,800,000	5,600,000
Other Liabilities [Member]
Income Tax Disclosure [Line Items]
Accrued income taxes	$ 10,700,000	$ 6,600,000