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Share-Based Compensation - Stock Options Activity (Details) - USD ($)
$ / shares in Units, $ in Millions
12 Months Ended
Sep. 30, 2023
Sep. 30, 2022
Sep. 30, 2021
Number of Options Outstanding      
Number of options outstanding, beginning (in shares) 1,199,345    
Number of options outstanding, granted (in shares) 81,750    
Number of options outstanding, exercised (in shares) (83,794)    
Number of options outstanding, forfeited (in shares) (19,849)    
Number of options outstanding, expired (in shares) (2,088)    
Number of options outstanding, ending (in shares) 1,175,364 1,199,345  
Number of options outstanding, exercisable (in shares) 387,810    
Weighted Average  Exercise Price      
Weighted average exercise price, beginning (in dollar per share) $ 50.46    
Weighted average  exercise price, granted (in dollar per share) 93.70    
Weighted average  exercise price, exercised (in dollar per share) 47.13    
Weighted average  exercise price, forfeited (in dollar per share) 65.10    
Weighted average  exercise price, expired (in dollar per share) 49.92    
Weighted average exercise price, ending (in dollar per share) 53.46 $ 50.46  
Weighted average  exercise price, exercisable (in dollar per share) 49.20    
Weighted Average Grant Date Fair Value      
Weighted average grant date fair value, beginning (in dollar per share) 14.57    
Weighted average grant date fair value, granted (in dollar per share) 33.21 22.82 $ 19.83
Weighted average grant date fair value, exercised (in dollar per share) 13.06    
Weighted average grant date fair value, forfeited (in dollar per share) 23.39    
Weighted average grant date fair value, expired (in dollar per share) 14.74    
Weighted average grant date fair value, ending (in dollar per share) 15.82 $ 14.57  
Weighted average grant date fair value, exercisable (in dollar per share) $ 13.61    
Stock Option Activity, Additional Disclosures      
Weighted average remaining term (in years) 3 years 9 months 18 days 4 years 6 months 14 days  
Weighted average remaining term (in years), exercisable 3 years 5 months 26 days    
Aggregate intrinsic value $ 51.1 $ 39.0  
Aggregate intrinsic value, exercisable $ 18.5