This is old topic but on my last test on one my API, cURL is faster and more stable. Sometimes file_get_contents on larger request need over 5 seconds when cURL need only from 1.4 to 1.9 seconds what is double faster.
I need to add one note on this that I just send GET and recive JSON content. If you setup cURL properly, you will have a great response. Just "tell" to cURL what you need to send and what you need to recive and that's it.
On your exampe I would like to do this setup:
$ch = curl_init('http://api.bitly.com/v3/shorten?login=user&apiKey=key&longUrl=url');
curl_setopt($ch, CURLOPT_RETURNTRANSFER, 1);
curl_setopt($ch, CURLOPT_SSL_VERIFYPEER, 0);
curl_setopt($ch, CURLOPT_HTTPAUTH, CURLAUTH_BASIC);
curl_setopt($ch, CURLOPT_CONNECTTIMEOUT, 5);
curl_setopt($ch, CURLOPT_TIMEOUT, 3);
curl_setopt($ch, CURLOPT_HTTPHEADER, array('Accept: application/json'));
$result = curl_exec($ch);
This request will return data in 0.10 second max