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Revenue Recognition - Disaggregation of Revenue (Details) - USD ($) $ in Thousands | 3 Months Ended | 12 Months Ended
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Sep. 30, 2019 | Jun. 30, 2019 | Mar. 31, 2019 | Dec. 31, 2018 | Sep. 30, 2018 | Jun. 30, 2018 | Mar. 31, 2018 | Dec. 31, 2017 | Sep. 30, 2019 | Sep. 30, 2018 | Sep. 30, 2017
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue | $ 754,966 | $ 730,710 | $ 736,520 | $ 664,619 | $ 558,446 | $ 597,855 | $ 612,787 | $ 623,148 | $ 2,886,815 | $ 2,392,236 | $ 2,450,961
Total U.S. state government agencies |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 1,166,937 |  |
New York State government agencies |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 362,724 |  |
Other U.S. state government agencies |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 804,213 |  |
United States Federal Government agencies |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 1,040,980 |  |
International government agencies |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 558,599 |  |
Other, including local municipalities and commercial customers |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 120,299 |  |
Performance-based |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 1,193,075 |  |
Cost-plus |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 1,088,541 |  |
Fixed price |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 441,146 |  |
Time and materials |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 164,053 |  |
U.S. Health and Human Services |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 1,176,488 |  |
U.S. Federal Services |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 1,111,197 |  |
Outside the U.S. |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 599,130 |  |
Program administration | U.S. Health and Human Services |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 883,772 |  |
Program administration | U.S. Federal Services |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 779,573 |  |
Program administration | Outside the U.S. |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 63,734 |  |
Assessments and appeals | U.S. Health and Human Services |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 136,109 |  |
Assessments and appeals | U.S. Federal Services |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 171,282 |  |
Assessments and appeals | Outside the U.S. |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 252,447 |  |
Workforce and children services | U.S. Health and Human Services |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 100,454 |  |
Workforce and children services | Outside the U.S. |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 272,801 |  |
Other | U.S. Health and Human Services |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 56,153 |  |
Other | Outside the U.S. |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | 10,148 |  |
Technology solutions | U.S. Federal Services |  |  |  |  |  |  |  |  |  |  |
Disaggregation of Revenue [Line Items] |  |  |  |  |  |  |  |  |  |  |
Revenue |  |  |  |  |  |  |  |  | $ 160,342 |  |

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