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Stock Based Compensation - Narrative (Details) - USD ($)
3 Months Ended 6 Months Ended
Jan. 18, 2022
Nov. 01, 2021
Feb. 28, 2022
Feb. 28, 2021
Feb. 28, 2022
Feb. 28, 2021
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]            
Stock-based compensation expense     $ 15,500,000 $ 11,000,000.0 $ 25,900,000 $ 22,300,000
Unrecognized compensation expense related to non-vested equity     $ 129,300,000   $ 129,300,000  
Weighted average period for recognition         3 years 2 months 12 days  
Exercise price (in USD per share) $ 428.71 $ 434.82        
Offering period         3 months  
Purchase price percentage         85.00%  
Maximum employee subscription rate     10.00%   10.00%  
Maximum contribution limit         $ 25,000  
Number of share purchased by employees (in shares)     8,232 9,528 17,417 18,797
Stock issued during period employee stock purchase plans weighted average price per share (in USD per share)     $ 351.06 $ 263.18 $ 340.41 $ 274.72
Capital shares reserved for future issuance (in shares)     121,539   121,539  
Restricted stock units            
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]            
Non-option award grants (in shares)         59,738  
PRSUs            
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]            
Non-option award grants (in shares)         30,704  
LTIP            
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]            
Stock options granted (in shares)         302,493  
Exercise price (in USD per share)         $ 434.70  
Number of shares available for grant (in shares)     4,700,000   4,700,000  
LTIP | Stock options            
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]            
Award vesting percentage   20.00%        
Award vesting period   5 years        
Expiration period   10 years        
LTIP | Restricted stock units            
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]            
Award vesting percentage         20.00%  
Award vesting period         5 years  
Weighted average grant date fair value (in USD per share)         $ 422.34  
Director Plan            
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]            
Stock options granted (in shares) 6,329          
Director Plan | Directors | Non-employee            
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]            
Award vesting percentage         100.00%  
Award vesting period         3 years  
Expiration period         7 years  
Number of shares available for grant (in shares)     227,348   227,348  
Director Plan | Restricted stock units | Directors            
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]            
Award vesting percentage 100.00%          
Non-option award grants (in shares) 1,629          
Weighted average grant date fair value (in USD per share) $ 425.49          
ESPP            
Share-based Compensation Arrangement by Share-based Payment Award [Line Items]            
Stock-based compensation expense     $ 500,000 $ 500,000 $ 1,100,000 $ 1,000,000.0