<s>[INST] How to evaluate $\lim_{n\to\infty}\int_0^n\frac{x^2+a^2}{x^4+b^2x^2+b^4}dx$ Evaluate this limit:
$$\lim_{n\to\infty}\int_0^n\dfrac{x^2+a^2}{x^4+b^2x^2+b^4}dx$$

I tried to simplify this fraction. I noticed that $x^4+b^2x^2+b^4$ can be written as
$$\dfrac{x^6-b^6}{x^2-b^2}$$
Then limit will be
$$\lim_{n\to\infty}\int_0^n\dfrac{\left(x^2+a^2\right)\left(x^2-b^2\right)}{x^6-b^6}dx$$
But what next? Is it possible to more simplify this fraction? Can we find indefinite integral of this fraction first?
 [/INST] Maple does the indefinite integral in terms of arctangents and logarithms.  So, yes, you can do it by partial fractions, where you first factor the denominator ... to factor it, complete the square, so it becomes a difference of squares
Then take limit to get the answer.  Maple says:
$$
\frac{(a^2+b^2)\pi}{b^3 2\sqrt{3}}
$$
assuming $b > 0$.
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