reCaptcha on custom HTML form

I am having trouble getting a reCaptcha validation to appear on a custom HTML form on one of my pages since the 3.6 update.

Can anyone suggest a plugin or method of coding the Captcha into the page that works in WP 3.6.1? The plugins I’ve seen only appear to work with default contact and comment forms – this is a custom form for payment processing.

The code that previously worked was:

<?php
  require_once('recaptchalib.php');
  $publickey = "##########";
  echo recaptcha_get_html($publickey);
?>

This PHP still displays the Captcha correctly when uploaded to the web server as a .php separate from WordPress, but no longer works on a WP page (even using the Insert PHP plugin).

This is for a cancer nonprofit, and people are spamming the form to test stolen credit cards without the Captcha there ??