Nature of Business Prior Year Restated (Details) - USD ($) $ / shares in Units, $ in Thousands		3 Months Ended		6 Months Ended
		Mar. 31, 2019	Mar. 31, 2018	Mar. 31, 2019	Mar. 31, 2018	Dec. 31, 2018	Sep. 30, 2018	Dec. 31, 2017	Sep. 30, 2017
Error Corrections and Prior Period Adjustments Restatement [Line Items]
Accounts receivable, net		$ 271,632		$ 271,632			$ 266,742
Deferred income taxes		13,678		13,678			13,805
Other assets		39,965		39,965			36,254
Other accrued liabilities		28,747		28,747			31,025
Deferred revenue		106,275		106,275			103,335
Stockholders’ equity		247,478	$ 414,823	247,478	$ 414,823	$ 227,895	287,437	$ 430,868	$ 466,183
Revenues		278,234	256,260	540,490	488,621
Cost of revenues	[1]	85,568	79,493	161,634	153,925
Selling, general and administrative	[1]	104,930	96,125	205,188	186,467
Provision for income taxes		6,593	7,993	3,742	5,992
Net income		33,381	31,169	73,388	64,048
Comprehensive income		$ 36,306	$ 40,294	$ 73,048	$ 76,244
Basic (in dollars per share)		$ 1.15	$ 1.04	$ 2.53	$ 2.13
Diluted (in dollars per share)		$ 1.10	$ 1.00	$ 2.42	$ 2.04
Deferred income taxes				$ 74	$ (3,152)
Changes in operating assets and liabilities					(34,041)
Previously Reported
Error Corrections and Prior Period Adjustments Restatement [Line Items]
Accounts receivable, net							208,865
Deferred income taxes							20,117
Other assets							12,431
Other accrued liabilities							30,457
Deferred revenue							52,215
Stockholders’ equity							263,737
Revenues			$ 257,858		493,179
Cost of revenues			78,519		152,878
Selling, general and administrative			97,057		187,353
Provision for income taxes			8,527		15,185
Net income			32,275		59,574
Comprehensive income			$ 41,400		$ 71,770
Basic (in dollars per share)			$ 1.08		$ 1.98
Diluted (in dollars per share)			$ 1.03		$ 1.90
Deferred income taxes					$ 6,041
Changes in operating assets and liabilities					(38,760)
Accounting Standards Update 2014-09 | Restatement Adjustment
Error Corrections and Prior Period Adjustments Restatement [Line Items]
Accounts receivable, net							57,877
Deferred income taxes							(6,312)
Other assets							23,823
Other accrued liabilities							568
Deferred revenue							51,120
Stockholders’ equity							$ 23,700
Revenues			$ (1,598)		(4,558)
Cost of revenues			974		1,047
Selling, general and administrative			(932)		(886)
Provision for income taxes			(534)		(9,193)
Net income			(1,106)		4,474
Comprehensive income			$ (1,106)		$ 4,474
Basic (in dollars per share)			$ (0.04)		$ 0.15
Diluted (in dollars per share)			$ (0.03)		$ 0.14
Deferred income taxes					$ (9,193)
Changes in operating assets and liabilities					$ 4,719

[1] Cost of revenues and selling, general and administrative expenses exclude the amortization of intangible assets. See Note 4.