Goodwill and Intangible Assets - Net - Summary of Finite-Lived Intangible Assets (Detail) - USD ($) $ in Thousands	6 Months Ended	12 Months Ended
	Mar. 31, 2023	Sep. 30, 2022
Finite Lived Intangible Assets [Line Items]
Gross Carrying Value	$ 720,068	$ 720,068
Accumulated Amortization	(497,690)	(474,233)
Net Carrying Value	$ 222,378	245,835
Proprietary Knowledge [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	10 years
Gross Carrying Value	$ 300,400	300,400
Accumulated Amortization	(244,899)	(236,024)
Net Carrying Value	$ 55,501	64,376
Trademarks [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	10 years
Gross Carrying Value	$ 230,240	230,240
Accumulated Amortization	(158,071)	(151,259)
Net Carrying Value	$ 72,169	78,981
Customer Relationships [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	12 years
Gross Carrying Value	$ 176,852	176,852
Accumulated Amortization	(85,258)	(78,015)
Net Carrying Value	91,594	98,837
Patents [Member]
Finite Lived Intangible Assets [Line Items]
Gross Carrying Value	8,500	8,500
Accumulated Amortization	(5,450)	(4,950)
Net Carrying Value	3,050	3,550
Other intangibles [Member]
Finite Lived Intangible Assets [Line Items]
Gross Carrying Value	4,076	4,076
Accumulated Amortization	(4,012)	(3,985)
Net Carrying Value	$ 64	$ 91
Minimum [Member] | Proprietary Knowledge [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	10 years	10 years
Minimum [Member] | Trademarks [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	5 years	5 years
Minimum [Member] | Customer Relationships [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	12 years	12 years
Minimum [Member] | Patents [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	9 years	9 years
Minimum [Member] | Other intangibles [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	3 years	3 years
Maximum [Member] | Proprietary Knowledge [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	15 years	15 years
Maximum [Member] | Trademarks [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	20 years	20 years
Maximum [Member] | Customer Relationships [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	19 years	19 years
Maximum [Member] | Patents [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	10 years	10 years
Maximum [Member] | Other intangibles [Member]
Finite Lived Intangible Assets [Line Items]
Acquired finite lived intangible asset useful life	15 years	15 years