Revenue from Contracts with Customers (Impact of ASC 606 on Balance Sheet) (Details) - USD ($) $ in Thousands	Sep. 30, 2019	Sep. 30, 2018	Sep. 30, 2017	Sep. 30, 2016
New Accounting Pronouncements or Change in Accounting Principle [Line Items]
Cash and cash equivalents	$ 269,579	$ 259,946
Short-term marketable securities	27,891	25,836
Accounts receivable, net of allowance for doubtful accounts of $744 and $607 at September 30, 2019 and 2018, respectively	372,743	129,297
Prepaid expenses	52,701	48,997
Other current assets	59,707	169,708
Total current assets	782,621	633,784
Property and equipment, net	105,531	80,613
Goodwill	1,238,179	1,182,457	$ 1,182,772
Acquired intangible assets, net	169,949	200,202
Long-term marketable securities	29,544	30,115
Deferred tax assets	198,634	165,566
Other assets	140,130	36,285
Total assets	2,664,588	2,329,022
Accounts payable	42,442	53,473
Accrued expenses and other current liabilities	104,028	74,388
Accrued compensation and benefits	88,769	101,784
Accrued income taxes	17,407	18,044
Deferred revenue	385,509	487,590
Total current liabilities	638,155	735,279
Long-term debt / Revolving credit facility	669,134	643,268
Deferred tax liabilities	41,683	5,589
Deferred revenue	11,123	11,852
Other liabilities	102,495	58,445
Total liabilities	1,462,590	1,454,433
Preferred stock	0	0
Common stock	1,149	1,180
Additional paid-in capital	1,502,949	1,558,403
Accumulated deficit	(191,390)	(599,409)
Accumulated other comprehensive loss	(110,710)	(85,585)
Total stockholders’ equity	1,201,998	874,589	$ 885,436	$ 842,666
Total liabilities and stockholders’ equity	2,664,588	$ 2,329,022
Calculated under Revenue Guidance in Effect before Topic 606 [Member]
New Accounting Pronouncements or Change in Accounting Principle [Line Items]
Cash and cash equivalents	269,579
Short-term marketable securities	27,891
Accounts receivable, net of allowance for doubtful accounts of $744 and $607 at September 30, 2019 and 2018, respectively	107,921
Prepaid expenses	54,384
Other current assets	199,513
Total current assets	659,288
Property and equipment, net	105,531
Goodwill	1,238,179
Acquired intangible assets, net	169,949
Long-term marketable securities	29,544
Deferred tax assets	233,026
Other assets	36,391
Total assets	2,471,908
Accounts payable	42,442
Accrued expenses and other current liabilities	78,007
Accrued compensation and benefits	88,769
Accrued income taxes	21,336
Deferred revenue	569,171
Total current liabilities	799,725
Long-term debt / Revolving credit facility	669,134
Deferred tax liabilities	14,644
Deferred revenue	9,577
Other liabilities	102,495
Total liabilities	1,595,575
Preferred stock	0
Common stock	1,149
Additional paid-in capital	1,502,949
Accumulated deficit	(524,169)
Accumulated other comprehensive loss	(103,596)
Total stockholders’ equity	876,333
Total liabilities and stockholders’ equity	$ 2,471,908