curl --include \
     --request PUT \
     --header 'Authorization: Bearer [TOKEN]' \
     --header 'Content-Type: application/vnd.contentful.management.v1+json' \
     --header 'X-Contentful-Content-Type: [CONTENT_TYPE_NAME]' \
     --data-binary '{
       "fields": {
          "internal": {
            "de-DE": "Ein Test"
          },
          "heading": {
            "de-DE": "Jawohl"
          },
          "level": {
            "de-DE": "H1"
          }
       }
     }' \
     https://api.contentful.com/spaces/[SPACE-ID]/environments/[ENV-ID]/entries/[ENTRY-ID]

The desired Content Type must be included in X-Contentful-Content-Type header. Note that the content is not published yet.