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Acquisitions - Additional Information (Details)
$ in Thousands
 12 Months Ended
Feb. 01, 2021
USD ($)
shares
Nov. 17, 2020
USD ($)
Sep. 30, 2023
USD ($)
business
Sep. 30, 2022
USD ($)
business
Sep. 30, 2021
USD ($)
business
Finite-Lived Intangible Assets [Line Items]          
Goodwill     $ 409,563 $ 353,639 $ 292,243
Software and Services          
Finite-Lived Intangible Assets [Line Items]          
Goodwill     287,613 234,553 173,157
Merchant Services          
Finite-Lived Intangible Assets [Line Items]          
Goodwill     $ 121,950 $ 119,086 $ 119,086
Celtic          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period     18 years    
Proceeds from issuance of long-term debt used to fund the acquisition     $ 85,000    
Acquisition-related costs     1,741    
Property and equipment     5,437    
Goodwill     43,695    
Other long-term liabilities     13,162    
Other Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Proceeds from issuance of long-term debt used to fund the acquisition     16,997    
Acquisition-related costs     $ 317    
Number of businesses acquired | business     2    
Total purchase consideration     $ 19,757    
Business combination, contingent consideration, liability     760    
Net working capital     159    
Property and equipment     374    
Goodwill     12,229    
Business acquisition, goodwill, expected tax deductible amount     2,864    
Other long-term liabilities     2,178    
Other Business Combinations | Class A Common Stock          
Finite-Lived Intangible Assets [Line Items]          
Common units issued to seller     2,000    
2022 Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period       15 years  
Proceeds from issuance of long-term debt used to fund the acquisition       $ 101,400  
Acquisition-related costs       $ 773  
Number of businesses acquired | business       3  
Total purchase consideration       $ 107,681  
Property and equipment       190  
Goodwill       61,347  
Other long-term liabilities       6,725  
Contingent cash consideration       $ 6,281  
Number of businesses acquired with tax deductible goodwill | business       2  
Potential additional consideration       $ 23,000  
2021 Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Number of businesses acquired | business         8
Property and equipment         $ 951
Goodwill         105,023
BIS          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period 19 years        
Proceeds from issuance of long-term debt used to fund the acquisition $ 52,500        
Acquisition-related costs 374        
Total purchase consideration 95,495        
Business combination, contingent consideration, liability 7,750        
Property and equipment         206
Goodwill         46,660
Potential additional consideration 16,000        
BIS | Class A Common Stock          
Finite-Lived Intangible Assets [Line Items]          
Common units issued to seller $ 35,245        
Business acquisition, equity interest issued or issuable, number of shares | shares 1,202,914        
ImageSoft, Inc.          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period   19 years      
Proceeds from issuance of long-term debt used to fund the acquisition   $ 40,000      
Acquisition-related costs   403      
Total purchase consideration   46,300      
Property and equipment         433
Goodwill         $ 22,408
Contingent cash consideration   6,300      
Potential additional consideration   $ 20,000      
Other          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period         16 years
Proceeds from issuance of long-term debt used to fund the acquisition         $ 57,000
Acquisition-related costs         $ 1,110
Number of businesses acquired | business         6
Total purchase consideration         $ 65,527
Property and equipment         312
Goodwill         35,955
Contingent cash consideration         8,527
Potential additional consideration         $ 50,200
Other | Software and Services          
Finite-Lived Intangible Assets [Line Items]          
Number of businesses acquired | business         5
Other | Merchant Services          
Finite-Lived Intangible Assets [Line Items]          
Number of businesses acquired | business         1
Residual buyouts          
Finite-Lived Intangible Assets [Line Items]          
Intangible assets acquired     $ 8,051 $ 52 $ 1,819
Estimated amortization period     8 years 8 years 8 years
Referral Agreements          
Finite-Lived Intangible Assets [Line Items]          
Intangible assets acquired     $ 420    
Estimated amortization period     5 years    
Acquired customer relationships | Celtic          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period     18 years    
Intangible assets     $ 33,800    
Acquired customer relationships | Other Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Intangible assets     $ 8,400    
Acquired customer relationships | Other Business Combinations | Minimum          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period     10 years    
Acquired customer relationships | Other Business Combinations | Maximum          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period     15 years    
Acquired customer relationships | 2022 Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Intangible assets       $ 41,090  
Acquired customer relationships | 2022 Business Combinations | Minimum          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period       10 years  
Acquired customer relationships | 2022 Business Combinations | Maximum          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period       19 years  
Acquired customer relationships | 2021 Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Intangible assets         $ 72,640
Acquired customer relationships | BIS          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period 19 years        
Intangible assets         32,300
Acquired customer relationships | ImageSoft, Inc.          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period   20 years      
Intangible assets         16,300
Acquired customer relationships | Other          
Finite-Lived Intangible Assets [Line Items]          
Intangible assets         $ 24,040
Acquired customer relationships | Other | Minimum          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period         10 years
Acquired customer relationships | Other | Maximum          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period         25 years
Trade name | Celtic          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period     5 years    
Intangible assets     $ 600    
Trade name | Other Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Intangible assets     $ 100    
Trade name | 2022 Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period       4 years  
Intangible assets       $ 1,550  
Trade name | 2021 Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Intangible assets         $ 2,640
Trade name | BIS          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period 5 years        
Intangible assets         700
Trade name | ImageSoft, Inc.          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period   5 years      
Intangible assets         $ 1,100
Trade name | Other          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period         4 years
Intangible assets         $ 840
Non-compete agreements | Celtic          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period     3 years    
Intangible assets     $ 200    
Non-compete agreements | 2021 Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Intangible assets         1,100
Non-compete agreements | BIS          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period 3 years        
Intangible assets         100
Non-compete agreements | ImageSoft, Inc.          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period   3 years      
Intangible assets         $ 610
Non-compete agreements | Other          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period         4 years
Intangible assets         $ 390
Capitalized software | Celtic          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period     10 years    
Intangible assets     $ 12,600    
Capitalized software | Other Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Intangible assets     $ 670    
Capitalized software | Other Business Combinations | Minimum          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period     7 years    
Capitalized software | Other Business Combinations | Maximum          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period     8 years    
Capitalized software | 2022 Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period       7 years  
Intangible assets       $ 9,790  
Capitalized software | 2021 Business Combinations          
Finite-Lived Intangible Assets [Line Items]          
Intangible assets         24,500
Capitalized software | BIS          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period 10 years        
Intangible assets         15,200
Capitalized software | ImageSoft, Inc.          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period   7 years      
Intangible assets         $ 5,200
Capitalized software | Other          
Finite-Lived Intangible Assets [Line Items]          
Estimated amortization period         7 years
Intangible assets         $ 4,100