Prepend www, redirect to https, and preserve full URL

If you are running at least 11.4, you can also achieve this using a Local Traffic Policy

• SOL15085: Overview of the Local Traffic Policies feature

It might look like this:

ltm policy www-redirect-append {
    controls { forwarding }
    requires { http }
    rules {
        rule-testsite.com {
            actions {
                0 {
                    http-reply
                    redirect
                    location "https://www.[getfield [HTTP::host] : 1][HTTP::uri]"
                }
            }
            conditions {
                0 {
                    http-host
                    values { testsite.com }
                }
            }
            ordinal 1
        }
        rule-default {
            actions {
                0 {
                    http-reply
                    redirect
                    location "https://[getfield [HTTP::host] : 1][HTTP::uri]"
                }
            }
            ordinal 2
        }
    }
    strategy first-match
}