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Segment and Geographic Information - Schedule of Segment Reporting Information (Details) - USD ($)
$ in Millions
 12 Months Ended
Sep. 30, 2024
Sep. 30, 2023
Sep. 30, 2022
Segment Reporting Information [Line Items]      
Total revenues $ 99,887.8 $ 60,856.1 $ 66,036.0
Operating revenues 3,436.2 2,914.1 2,107.4
Net operating revenues (loss) 1,767.2 1,621.0 1,475.9
Net contribution 1,401.6 1,273.6 1,121.9
Segment income 885.6 763.5 675.7
Net operating loss within Corporate (64.7) (62.9) (59.5)
Overhead costs and expenses (466.8) (400.9) (339.0)
Gain on acquisitions and other gains, net within Corporate 0.0 23.3 0.0
Income before tax 354.1 323.0 277.2
Assets 27,466.3 21,938.7 19,859.6
Corporate      
Segment Reporting Information [Line Items]      
Total revenues 46.9 31.7 7.8
Operating revenues 46.9 31.7 7.8
Net operating revenues (loss) (64.7) (62.9) (59.5)
Assets 1,123.5 812.3 746.3
Eliminations      
Segment Reporting Information [Line Items]      
Total revenues (49.3) (39.5) (23.0)
Operating revenues (49.3) (39.5) (23.0)
Commercial      
Segment Reporting Information [Line Items]      
Net contribution 541.8 544.9 415.3
Segment income 387.7 390.7 288.3
Commercial | Operating Segments      
Segment Reporting Information [Line Items]      
Total revenues 97,278.3 58,249.2 63,743.2
Operating revenues 871.9 862.7 692.1
Net operating revenues (loss) 715.9 721.3 586.5
Assets 5,387.0 4,676.3 5,931.0
Institutional      
Segment Reporting Information [Line Items]      
Net contribution 430.3 351.5 295.1
Segment income 266.0 217.9 174.6
Institutional | Operating Segments      
Segment Reporting Information [Line Items]      
Total revenues 1,962.1 1,513.6 831.8
Operating revenues 1,962.1 1,513.6 831.8
Net operating revenues (loss) 630.4 532.0 483.5
Assets 19,492.9 15,059.3 11,687.1
Self-Directed/Retail      
Segment Reporting Information [Line Items]      
Net contribution 267.0 212.7 280.3
Segment income 119.3 45.8 115.4
Self-Directed/Retail | Operating Segments      
Segment Reporting Information [Line Items]      
Total revenues 440.2 888.5 1,304.2
Operating revenues 395.0 333.0 426.7
Net operating revenues (loss) 286.1 227.3 302.9
Assets 1,024.1 1,014.2 971.2
Payments      
Segment Reporting Information [Line Items]      
Net contribution 162.5 164.5 131.2
Segment income 112.6 109.1 97.4
Payments | Operating Segments      
Segment Reporting Information [Line Items]      
Total revenues 209.6 212.6 172.0
Operating revenues 209.6 212.6 172.0
Net operating revenues (loss) 199.5 203.3 162.5
Assets $ 438.8 $ 376.6 $ 524.0