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Income Taxes - Additional Information (Details) - USD ($)
 3 Months Ended 12 Months Ended
Sep. 30, 2016
Sep. 30, 2021
Sep. 30, 2020
Sep. 30, 2019
Sep. 30, 2018
Income Tax Disclosure [Line Items]          
Income taxes payable   $ 15,700,000 $ 15,400,000    
Accrued income taxes   5,000,000.0 7,000,000.0    
Prepaid income taxes   15,400,000 17,300,000    
Income tax payments   56,000,000.0 52,600,000 $ 38,900,000  
Non-cash federal   109,400,000      
State tax benefits   24,800,000      
Foreign tax credits   36,287,000 8,074,000    
Gross deferred amortization carryforward   1,084,600,000      
Valuation allowance   52,085,000 205,423,000 177,663,000 $ 141,950,000
Interest Expense   50,478,000 76,428,000 43,047,000  
Penalty expense   2,000,000.0 0 0  
Interest expense related to income tax accruals   700,000 600,000    
Accrued tax penalties   0 0 0  
Income tax provision upon recognition of unrecognized tax benefit   21,200,000      
Unrecognized tax benefits, increase in valuation allowance upon recognition   4,500,000      
Potential decrease in unrecognized tax benefits   1,000,000      
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   177,289,000 115,149,000 86,400,000  
Tax benefit recognition related to stock based compensation   39,900,000 13,400,000 16,600,000  
Provision (benefit) for income taxes   (85,168,000) 4,011,000 47,760,000  
Income Tax Expense          
Income Tax Disclosure [Line Items]          
Interest Expense   2,200,000 300,000 100,000  
Domestic Country          
Income Tax Disclosure [Line Items]          
Net operating loss carryforwards   190,300,000      
Valuation allowance   17,700,000      
Domestic Country | Expire in 2023 to 2036          
Income Tax Disclosure [Line Items]          
Credit carryforwards   54,100,000      
Domestic Country | Expire in 2023 to 2036          
Income Tax Disclosure [Line Items]          
Net operating loss carryforwards   32,500,000      
Domestic Country | Not Expire          
Income Tax Disclosure [Line Items]          
Net operating loss carryforwards   157,800,000      
State - Massachusetts          
Income Tax Disclosure [Line Items]          
Valuation allowance   17,700,000      
State - Massachusetts | Research Tax Credit Carryforward          
Income Tax Disclosure [Line Items]          
Credit carryforwards   24,200,000      
Foreign Country          
Income Tax Disclosure [Line Items]          
Net operating loss carryforwards   49,800,000      
Credit carryforwards   $ 36,300,000      
Credit carryforwards expiration   2025      
Foreign tax credits   $ 4,700,000      
Valuation allowance   $ 34,400,000      
Foreign Country | Minimum          
Income Tax Disclosure [Line Items]          
Non-U.S. tax credit carryforwards expiration   2030      
Foreign Country | Maximum          
Income Tax Disclosure [Line Items]          
Non-U.S. tax credit carryforwards expiration   2040      
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 $ 12,000,000 20,000,000      
Windfall Tax Benefit          
Income Tax Disclosure [Line Items]          
Provision (benefit) for income taxes   9,900,000 1,300,000 $ 6,700,000  
Other Current Liabilities          
Income Tax Disclosure [Line Items]          
Accrued income taxes   800,000 1,000,000.0    
Other Liabilities [Member]          
Income Tax Disclosure [Line Items]          
Accrued income taxes   $ 9,900,000 $ 7,400,000