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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
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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 |  |

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