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Income Taxes (Details) - USD ($)
 12 Months Ended
Aug. 31, 2019
Aug. 31, 2018
Aug. 31, 2017
Effective Income Tax Rate Reconciliation, Amount [Abstract]      
Federal income tax computed at statutory rate $ 89,200,000 $ 109,400,000 $ 172,400,000
State income tax, net of federal income tax benefit 12,200,000 11,500,000 12,200,000
Foreign permanent differences and rate differential 2,100,000 (2,000,000.0) (1,600,000)
Discrete income tax benefits of the TCJA (2,200,000) (34,600,000) 0
Research and development tax credits (18,100,000) (3,300,000) (3,000,000.0)
Unrecognized tax benefits 12,200,000 400,000 800,000
Other, net (900,000) (5,100,000) (9,900,000)
Total provision for income taxes 94,500,000 76,300,000 170,900,000
Deferred income tax liabilities:      
Depreciation (22,000,000.0) (15,000,000.0)  
Goodwill and intangibles (149,600,000) (151,200,000)  
Other liabilities (2,800,000) (2,300,000)  
Total deferred income tax liabilities (174,400,000) (168,500,000)  
Deferred income tax assets:      
Self-insurance 2,600,000 2,600,000  
Pension 22,700,000 18,100,000  
Deferred compensation 20,500,000 23,700,000  
Net operating losses 6,200,000 6,200,000  
Other accruals not yet deductible 26,900,000 24,900,000  
Other assets 9,700,000 7,000,000.0  
Total deferred income tax assets 88,600,000 82,500,000  
Valuation allowance (4,600,000) (3,600,000)  
Net deferred income tax liabilities (90,400,000) (89,600,000)  
Discrete tax benefit 36,800,000    
Provisional benefit to decrease Company's deferred income taxes to a revised statutory federal rate 32,500,000    
Current estimated benefit for transition tax on unremitted foreign earnings 4,300,000    
Undistributed earnings and original investments in foreign subsidiaries 107,700,000    
Deferred income tax liability for certain foreign withholding taxes and U.S. state taxes 600,000    
State tax credit carryforwards 2,200,000    
Unrecognized tax benefits 16,600,000 4,400,000 $ 6,000,000.0
Unrecognized tax benefits that if recognized would affect annual effective tax rate 15,900,000 3,800,000  
Total accrued interest 1,000,000.0 $ 500,000  
Accruals related to income tax penalties 0    
Internal Revenue Service (IRS)      
Deferred income tax assets:      
Operating loss carryforwards 32,900,000    
State and Local Jurisdiction      
Deferred income tax assets:      
Operating loss carryforwards 20,300,000    
Foreign Tax Authority      
Deferred income tax assets:      
Operating loss carryforwards $ 1,800,000