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Revenue Recognition - Disaggregation of Revenue (Details) - USD ($) $ in Thousands | 3 Months Ended | 9 Months Ended
--- | --- | ---
Jun. 30, 2021 | Jun. 30, 2020 | Jun. 30, 2021 | Jun. 30, 2020
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | $ 1,243,520 | $ 901,337 | $ 3,148,354 | $ 2,537,701
United States |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 1,053,940 | 787,092 | 2,622,469 | 2,168,033
United Kingdom |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 74,109 | 53,364 | 211,777 | 190,088
Australia |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 65,283 | 38,415 | 192,161 | 100,390
Rest of world |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 50,188 | 22,466 | 121,947 | 79,190
Total U.S. state government agencies |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 431,987 | 344,748 | 1,265,798 | 969,274
New York State government agencies |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 102,195 | 79,140 | 289,462 | 276,585
Other U.S. state government agencies |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 329,792 | 265,608 | 976,336 | 692,689
U.S. Federal Government agencies |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 594,771 | 429,031 | 1,288,213 | 1,155,773
International government agencies |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 180,049 | 107,353 | 499,091 | 345,629
Other, including local municipalities and commercial customers |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 36,713 | 20,205 | 95,252 | 67,025
Performance-based |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 389,800 | 295,650 | 1,033,509 | 864,077
Cost-plus |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 282,808 | 422,641 | 953,373 | 1,184,425
Fixed price |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 146,175 | 132,535 | 413,296 | 352,255
Time and materials |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 424,737 | 50,511 | 748,176 | 136,944
U.S. Services |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 436,338 | 336,950 | 1,269,487 | 957,929
U.S. Federal Services |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 617,601 | 450,143 | 1,352,982 | 1,210,105
Outside the U.S. |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 189,581 | 114,244 | 525,885 | 369,667
Program administration | U.S. Services |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 325,755 | 251,616 | 964,981 | 724,960
Program administration | U.S. Federal Services |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 451,594 | 362,973 | 1,005,706 | 949,071
Program administration | Outside the U.S. |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 34,296 | 15,544 | 80,284 | 49,583
Technology solutions | U.S. Federal Services |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 89,559 | 42,101 | 187,006 | 130,172
Assessments and appeals | U.S. Services |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 33,529 | 40,875 | 101,561 | 104,621
Assessments and appeals | U.S. Federal Services |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 76,448 | 45,069 | 160,270 | 130,862
Assessments and appeals | Outside the U.S. |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 58,664 | 42,468 | 169,753 | 167,397
Workforce and children services | U.S. Services |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 64,240 | 32,531 | 167,106 | 90,651
Workforce and children services | Outside the U.S. |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 94,310 | 55,046 | 269,743 | 146,968
Other | U.S. Services |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | 12,814 | 11,928 | 35,839 | 37,697
Other | Outside the U.S. |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |
Revenue | $ 2,311 | $ 1,186 | $ 6,105 | $ 5,719

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