Like you said, you would need to use partial fractions:
You can verify that:
So our new integral is:
For the second term, you're going to have to complete the square and use a trigonometric substitution. See how far you can go with that one. It'll probably be pretty long and tedious.
For the second integral, multiply both top and bottom by .
Now use a few trigonometric identities to get your integral to become:
which are standard ones.