$\int \frac{d x}{x\left(x^5+3\right)}$ is equal to

$\frac{1}{15} \log \left|\frac{x^5}{x^5+3}\right|+C$

$\int \frac{1}{x\left(x^5+3\right)}dx$

= $\int \frac{1}{x^6\left(1+3x^{-5}\right)}dx$

let $y = 1 + 3x^{-5}$

⇒  $dy = \frac{-15}{x^6}dx$

⇒  $\frac{-1~dy}{15} = \frac{1}{x^6}dx$

⇒  $I = \frac{-1}{15} \int \frac{dy}{y} = \frac{-1}{15} \log |y| + C$

$I = \frac{-1}{15} \log |1+3x^{-5}| + C$

= $\frac{-1}{15} \log |\frac{x^5+3}{x^5}| + C$

= $\frac{1}{15} \log |\frac{x^5}{x^5+3}| + C$