The captcha does not appear in google chrome when using https, due to the fact the call to the script is not secure (https).

In your call to the recaptcha_get_html() function, on line 62 of dmxReCAPTCHA.php you specify the last parameter ($use_ssl) as false.
It would be good to have this function automatically check if the page is served via https, and set this true if needed.

Thanks,
Tom