Add a test and fix for the request body being closed twice.

Fixes #19186

Change-Id: I1e35ad4aebfef68e6099c1dba7986883afdef4d7
Reviewed-on: https://go-review.googlesource.com/37298Reviewed-by: Emmanuel Odeke <emm.odeke@gmail.com>
Reviewed-by: Brad Fitzpatrick <bradfitz@golang.org>
Run-TryBot: Emmanuel Odeke <emm.odeke@gmail.com>
TryBot-Result: Gobot Gobot <gobot@golang.org>