Why does PHP refuse to enable cURL on Windows

I am using PHP 5.5.25 with Apache 2.4 on Windows 7 x64 and I am unable to activate the cURL module. I have looked around and tried all I could think of. Please assist:

• In php.ini, the line extension=php_curl.dll is active and the file php_curl.dll is present in the extensions directory C:\php\ext
• In php.ini when I set extension_dir = ext, none of the extensions load. I get several messages when Apache starts, similar to Unable to load dynamic libraryext\php_openssl.dll- The specified module could not be found.
• When I use the full path and set extension_dir = C:\php\ext, all the extensions load fine, except for cURL. I get the error: Unable to load dynamic libraryC:\php\ext\php_curl.dll- The specified module could not be found.
• I have tried renaming the extension to php_curl.new.dll and adjusting php.ini but I get an error message about the new filename. I have also downloaded a fresh new copy of the DLL from windows.php.net, but that made no difference.
• I have checked the file permissions for php_curl.dll (Right-click on the file >> Properties >> Security tab) and they are the same as the permissions for extensions that load successfully
• I have copied and pasted libeay32.dll and ssleay32.dll from the PHP bin directory to the System32 and SysWOW64 directories as instructed by a response to this question
• I am certain that I am editing the right php.ini since the PHP startup error messages changed when I changed the extension_dir value from ext to C:\php\ext as I explained above.
• I have made sure to restart the Apache server between php.ini configuration changes.
• If in a PHP script I execute var_dump(file_exists('C:\php\ext\php_curl.dll'));, I get boolean True so PHP can see the file!

What else could explain why the cURL module is not enabled?

I think you'll need libssh2.dll in your PATH too.