[DOCS] Move async search API docs after search request body API docs (#56244)

docs/reference/search.asciidoc

@@ -152,10 +152,10 @@ high). This default value is `5`.
 
 include::search/search.asciidoc[]
 
-include::search/async-search.asciidoc[]
-
 include::search/request-body.asciidoc[]
 
+include::search/async-search.asciidoc[]
+
 include::search/search-template.asciidoc[]
 
 include::search/search-shards.asciidoc[]