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I wanted to find out if there is an easy way to evaluate $$\displaystyle\int\frac{1}{x^{11}+4x^6}dx.$$

I substituted $u=x^5$ and then used partial fractions, but maybe there is a simpler way to find this.

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Let $t=\frac{1}{x}$, so then $x=\frac{1}{t}$ and $dx=-\frac{1}{t^2}dt$.

Then $\displaystyle\int\frac{1}{x^{11}+4x^6}dx=\int\frac{1}{\frac{1}{t^{11}}+\frac{4}{t^6}}\left(-\frac{1}{t^2}\right)dt=-\int\frac{t^9}{1+4t^5}dt$

$=-\int\big(\frac{1}{4}t^4-\frac{\frac{1}{4}t^4}{1+4t^5}\big)dt=-\frac{1}{20}t^5+\frac{1}{80}\ln\vert1+4t^5\vert+C=-\frac{1}{20x^5}+\frac{1}{80}\ln\vert1+\frac{4}{x^5}\vert+C$

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Hint: $$ \begin{align} \int\frac{dx}{x^{11}+4x^6}&=\int\frac{x^4dx}{x^{15}+4x^{10}}\\ &=\frac14\int\frac{du}{u^3+4u^2}\,\,\,\,\,,u=x^5\\ &=\frac{1}{16}\int\bigg[\frac{1}{u^2}-\frac{1}{4u}+\frac{1}{4(4+u)}\bigg]du \end{align} $$

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After using partial fractions, try the substitution $u=x^5+4$ to deal with the first integral in the expression \begin{equation*} \frac{1}{16}\int\frac{x^4}{x^5+4}dx-\frac{1}{16}\int\frac{1}{x}dx+\frac{1}{4}\int\frac{1}{x^6}dx \end{equation*} which is what you get when you use partial fractions on your integrand.

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$$\int\frac{1}{x^{11}+4x^6}dx=$$ $$\int \left(\frac{1}{4x^6}+\frac{x^4}{16(x^5+4)}-\frac{1}{16x}\right)dx=$$ $$\frac{1}{16}\int \frac{x^4}{x^5+4}dx-\frac{1}{16}\int \frac{1}{x}dx+\frac{1}{4}\int \frac{1}{x^6}dx=$$ $$\frac{1}{80}\int \frac{1}{u}du-\frac{1}{16}\int\frac{1}{x}dx+\frac{1}{4}\int\frac{1}{x^6}dx=$$ $$\frac{\ln(u)}{80}-\frac{1}{16}\int\frac{1}{x}dx+\frac{1}{4}\int\frac{1}{x^6}dx=$$ $$\frac{\ln(u)}{80}-\frac{\ln(x)}{16}+\frac{1}{4}\int\frac{1}{x^6}dx=$$ $$\frac{\ln(u)}{80}-\frac{1}{20x^5}-\frac{\ln(x)}{16}+C=$$ $$\frac{1}{80}\left(-\frac{4}{x^5}+\ln(x^5+4)-5\ln(x)\right)+C$$

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