$\int \frac{8{\mathrm{x}}^{43}+13{\mathrm{x}}^{38}}{{({\mathrm{x}}^{13}+{\mathrm{x}}^5+1)}^4}\mathrm{dx}=$

$\begin{array}{l}\int \frac{8x^{43}+13x^{38}}{{(x^{13}+x^5+1)}^4}dx\\ \int \frac{8x^{-9}+13x^{-14}}{{(1+x^{-8}+x^{-13})}^4}dx\\ \text{ Let }1+x^{-8}+x^{-13}=t\\ (-8x^{-9}-13x^{-14})dx=dt\end{array}$

$$\begin{gathered} \therefore \int -\frac{\mathrm{dt}}{{\mathrm{t}}^4}=\frac{1}{3{\mathrm{t}}^3}+\mathrm{C} \\ =\frac{1}{3{(1+{\mathrm{x}}^{-8}+{\mathrm{x}}^{-13})}^3}+\mathrm{C} \\ =\frac{{\mathrm{x}}^{39}}{3{({\mathrm{x}}^{13}+{\mathrm{x}}^5+1)}^{13}}+\mathrm{C} \end{gathered}$$