# Integrate: 5x^4+4x^5 / (x^5+x+1)^2

Solution:

$I=\int \frac{5x^{4}+4x^{5}}{(x^{5}+x+1)^{2}}dx \\ \\ \Rightarrow \hspace{1cm} I=\int \frac{5x^{4}+4x^{5}}{x^{10}(1+\frac{1}{x^{4}}+\frac{1}{x^{5}})^{2}}dx\\ \\ \Rightarrow \hspace{1cm}I=\int \frac{\frac{5}{x^{6}}+\frac{4}{x^{5}}}{(1+\frac{1}{x^{4}}+\frac{1}{x^{5}})^{2}}dx\\ \\ \Rightarrow \hspace{1cm} Let \hspace{1cm}1+\frac{1}{x^{4}}+\frac{1}{x^{5}}=t\Rightarrow -\frac{4}{x^{5}}-\frac{5}{x^{6}}dx=dt$

$\\ \\ \Rightarrow \hspace{1cm}I=-\int \frac{1}{t^{2}}dt=\frac{1}{t}+c\\ \\ \Rightarrow \hspace{1cm}I=\frac{x^{5}}{x^{5}+x+1}+c$

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