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Goodwill and Intangible Assets - Narrative (Details) - USD ($)
$ in Millions
3 Months Ended 9 Months Ended
May 31, 2024
May 31, 2023
May 31, 2024
May 31, 2023
Goodwill and Intangible Assets Disclosure [Abstract]        
Amortization of intangible assets $ 10.0 $ 9.2 $ 29.9 $ 32.1