Chapter 7 - Section 7.4 - Integration of Rational Functions by Partial Fractions. - 7.4 Exercises - Page 515: 38

$\int$ $\frac{x^{4}+3x^2+1}{x^5+5x^3+5x}$ $dx =$ $\frac{1}{5} ln |x^5+5x^3+5x|+c$

$let $ $u=x^5+5x^3+5x$ $ du = 5(x^4 + 3x^2+1)dx$ $So, $ $ dx = \frac{du}{5(x^4 + 3x^2+1)}$ $Then$ $we $ $ have $ $:$ $\frac{1}{5} \int \frac{1}{u} du$ $so$ $we$ $get, $ $\frac{1}{5} ln |u| +c$ $which$ $is, $ $\frac{1}{5} ln |x^5+5x^3+5x| +c$