Revenue Recognition - Disaggregation of Revenue (Details) - USD ($) $ in Thousands	3 Months Ended								12 Months Ended
	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