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I was wondering if someone would be kind enough to walk me through the logic used in solving the following integral. I have been to class, and have read the section (9.4 of Swokowski's Classic), and have studied the answer in the solution manual, but I can't quite seem to make sense of the rules posed (p.474, Swokowski's Classic) for the decomposition.

$$ \int\frac{x^2+3x+1}{x^4+5x^2+4}dx $$

Factors to:

$$ \int\frac{x^2+3x+1}{(x^2+4)(x^2+1)}dx $$

And this is where I get completely lost. I can do simple ones, such as

$$ \int\frac{x+16}{x^2+2x-8} $$

where they reduce to

$$ \frac{A}{x+4} + \frac{B}{x-2} $$

But the solution manual suggests that A and B should be Ax+B and Cx+D, referring to the aforementioned rule, and I'm quite confused.

Thank you very much!

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