In the video the cranking speed sounds normal. But the only real way to know is actually measure the cranking RPM and monitor the cranking battery voltage. Cranking voltage rule of thumb for a 12 volt battery is greater that 9.6 volts (about 10 volts) the battery is ok.
Battery capacity tests will also help when diagnosing battery issues. Kragen/O'Riely auto parts stores will often capacity test the battery for free. Just make sure they input the CCA (cold cranking amp) rating for your battery correctly. If the tester is set-up with a higher (incorrect) rating they will be selling you a battery.
If you have a battery charger, fully charging the battery before testing will improve the results.