$\int \frac{3x^2-4x^9}{2x^7+1+x^{14}}dx=$

$\int \frac{3x^2-4x^9}{2x^7+1+x^{14}}dx$

$=-\int \frac{x^6(4x^3-3x^{-4})}{x^6(x^8+2x+x^{-6})}dx$

$\begin{array}{r}=-\int \frac{4x^3-3x^{-4}}{{\left(x^4+\frac{1}{x^3}\right)}^2}dx\\ \text{ Let }{x}^4+\frac{1}{x^3}=t\\ \left(4x^3-3x^{-4}\right)dx=dt\\ \begin{array}{l}\int \frac{3x^2-4x^9}{2x^7+1+x^{14}}dx=-\int \frac{1}{t^2}dt\\ =\frac{1}{t}+c\\ =\frac{1}{x^4+\frac{1}{x^3}}+c\end{array}\end{array}$

$=\frac{x^3}{x^7+1}+c$