Answer
$\frac{1}{5}\ln|x^5+5x^3+5x|+C$
Work Step by Step
$\int\frac{x^4+3x^2+1}{x^5+5x^3+5x}dx$ = $\int\frac{x^4+3x^2+1}{x(x^4+5x^2+5)}dx$ = $\int\left(\frac{A}{x}+\frac{Bx^3+Cx^2+Dx+E}{x^4+5x^2+5}\right)dx$
$x^4+3x^2+1$ = $A(x^4+5x^2+5)+(Bx^3+Cx^2+Dx+E)x$
$x^4+3x^2+1$ = $A(x^4+5x^2+5)+(Bx^4+Cx^3+Dx^2+Ex)$
$1$ = $A+B$
$0$ = $C$
$3$ = $5A+D$
$0$ = $E$
$1$ = $5A$
$A$ = $\frac{1}{5}$, $B$ = $\frac{4}{5}$, $C$ = $0$, $D$ = $2$, $E$ = $0$
so
$\int\frac{x^4+3x^2+1}{x^5+5x^3+5x}dx$ = $\int{\frac{1}{5}}\left(\frac{1}{x}+\frac{4x^3+10x}{x^4+5x^2+5}\right)dx$ = $\frac{1}{5}(\ln|x|+\ln|x^4+5x^2+5|)+C_1$ = $\frac{1}{5}\ln|x^5+5x^3+5x|+C$
