Apex test classes for the BitlyService. The .java extension is just for syntax highlighting, save them as .cls in your project.

BitlyHttpCalloutMock.java

@IsTest
public class BitlyHttpCalloutMock implements HttpCalloutMock {

    public HttpResponse respond(HttpRequest req) {

        String endpoint = req.getEndpoint();

        if ( endpoint.contains('/oauth/access_token') ) {
            return buildOAuthResponse( req );
        } else if ( endpoint.contains('/v4/shorten') ) {
            return buildShortenResponse( req );
        }

        return null;
    }
    
    private HttpResponse buildOAuthResponse(HttpRequest req) {

        HttpResponse res = new HttpResponse();

        res.setBody('123');
        res.setStatusCode(200);

        return res;
    }

    private HttpResponse buildShortenResponse(HttpRequest req) {

        HttpResponse res = new HttpResponse();

        // See https://dev.bitly.com/api-reference#createBitlink
        // for full details of what a response could include.
        res.setBody(JSON.serializePretty(new Map<String, Object>{
            'link' => 'https://sforce.co/3rUWGZ8'
        }));
        res.setStatusCode(200);

        return res;
    }

}

BitlyServiceTest.java

@IsTest
private class BitlyServiceTest {

    @IsTest
    static void test_shorten() {

        Test.startTest();

        Test.setMock(HttpCalloutMock.class, new BitlyHttpCalloutMock());

        BitlyService service = new BitlyService();

        String longURL = 'https://this-is-a-test.example.com';
        String shortURL = service.shorten(longURL);

        Test.stopTest();

        System.assertEquals('https://sforce.co/3rUWGZ8', shortURL);

    }

}

BitlyShortenURLInvocableTest.java

@IsTest
private class BitlyShortenURLInvocableTest {

    @IsTest
    static void test_shorten() {

        // If there is an active process on Case object that invokes the
        // BitlyShortenURLInvocable as outlined in my blog post
        // https://douglascayers.wordpress.com/2015/10/21/salesforce-create-short-urls-with-bitly-process-builder-and-apex/
        // then simply inserting the record will cause the code to run and ensure code coverage.
        // Otherwise, the other test method is sufficient.

        Test.startTest();

        Test.setMock(HttpCalloutMock.class, new BitlyHttpCalloutMock());

        Case caseObj = new Case();

        insert caseObj;

        Test.stopTest();

        caseObj = [ SELECT Id, Short_URL__c FROM Case WHERE Id = :caseObj.Id LIMIT 1 ];

        System.assert( String.isNotBlank( caseObj.Short_URL__c ) );

    }

    @IsTest
    static void test_shorten2() {

        // This test just ensures the invocable class will run.
        // If we were to have an active process or trigger then
        // inserting the record would be sufficient, as in the above test method.
        // If not, then this method will be sufficient for code coverage.
        //
        // However, do note that trying to do both in the same test will fail.
        // Trying to insert a record and call the invocable class will fail
        // because the invocable class calls a @Future method that makes an HttpCallout.
        // DML operations don't mix with HttpCallouts due to uncommitted work:
        // "System.CalloutException: You have uncommitted work pending. Please commit or rollback before calling out.""

        Test.startTest();

        Test.setMock(HttpCalloutMock.class, new BitlyHttpCalloutMock());

        BitlyShortenURLInvocable.shorten( new String[] {} );

        Test.stopTest();

    }

}

Code updated to use Bitly's API v4