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.

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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.