<s>[INST] Integrating $\int \frac{3}{(x^2 +5)^2}dx$ by parts Integrating $$\int \frac{3}{(x^2 +5)^2}dx$$
After removing the constant, it is basically integrating $\frac{1}{x^4+10x^2+25}$. I only have learnt up to integrating $\frac{1}{ax^2 + bx +c}$ with the highest power of $x$ is 2. And this cannot be broken up into partial fractions too. What happens when the power of $x$ is 4?
My thoughts are to do integration by parts (product rule)
$$\int u v' dx = uv - \int u' v dx$$
But I am unclear what do substitute $u$ and $v$ to
 [/INST] Integrate by parts as follows
\begin{align}
\int \frac{3}{(x^2 +5)^2}dx &=\int \frac3{10x}d\left(\frac{x^2}{x^2+5}\right)\\
&= \frac{3x}{10(x^2+5)}+\frac3{10}\int\frac{1}{x^2+5}dx\\
 &= \frac{3x}{10(x^2+5)}+\frac3{10\sqrt5}\tan^{-1}\frac x{\sqrt5}+C
\end{align}
</s>