The general consensus is that you do not (should not) need to dispose of HttpClient.
Many people who are intimately involved in the way it works have stated this.
See Darrel Miller’s blog post and a related SO post: HttpClient crawling results in memory leak for reference.
I’d also strongly suggest that you read the HttpClient chapter from Designing Evolvable Web APIs with ASP.NET for context on what is going on under the hood, particularly the “Lifecycle” section quoted here:
Although HttpClient does indirectly implement the IDisposable
interface, the standard usage of HttpClient is not to dispose of it
after every request. The HttpClient object is intended to live for as
long as your application needs to make HTTP requests. Having an object
exist across multiple requests enables a place for setting
DefaultRequestHeaders and prevents you from having to re-specify
things like CredentialCache and CookieContainer on every request as
was necessary with HttpWebRequest.
Or even open up DotPeek.