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Revenue Recognition - Disaggregation of Revenue (Details) - USD ($) $ in Thousands | 3 Months Ended
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Dec. 31, 2018 | Dec. 31, 2017
Disaggregation of Revenue [Line Items] |  |
Revenue | $ 664,619 | $ 623,148
United States of America |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 511,200 |
United Kingdom |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 73,418 |
Australia |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 53,373 |
Rest of world |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 26,628 |
Total U.S. state government agencies |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 290,614 |
New York State government agencies |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 91,712 |
Other U.S. state government agencies |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 198,902 |
United States Federal Government agencies |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 198,278 |
International government agencies |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 142,781 |
Other, including local municipalities and commercial customers |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 32,946 |
Performance-based |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 312,887 |
Cost-plus |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 175,298 |
Fixed price |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 147,151 |
Time and materials |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 29,283 |
Program administration | U.S. Health & Human Services |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 218,973 |
Program administration | U.S. Federal Services |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 140,121 |
Program administration | Outside the U.S. |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 15,320 |
Assessments and appeals | U.S. Health & Human Services |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 37,221 |
Assessments and appeals | U.S. Federal Services |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 37,983 |
Assessments and appeals | Outside the U.S. |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 62,310 |
Workforce and children services | U.S. Health & Human Services |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 23,903 |
Workforce and children services | Outside the U.S. |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 73,278 |
Other | U.S. Health & Human Services |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 14,116 |
Other | Outside the U.S. |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | 2,511 |
Technology solutions | U.S. Federal Services |  |
Disaggregation of Revenue [Line Items] |  |
Revenue | $ 38,883 |

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