$I=\int\limits_{1}^{3}\frac{dx}{x^{9}+3x^{5}}=\int\limits_{1}^{3}\frac{dx}{x^{5}(x^{4}+3)}=\int\limits_{1}^{3}\frac{x^{3}dx}{x^{8}(x^{4}+3)}$
đặt $x^{4}=t\Rightarrow dt=4x^{3}dx$
$I=\int\limits_{1}^{81}\frac{dt}{4t^{2}(t+3)}=\int\limits_{1}^{81}[\frac{1}{12t^{2}}-\frac{1}{36t}+\frac{1}{36(t+3)}]dt$
$=[-\frac{1}{12t}-\frac{ln|t|}{36}+\frac{ln|t+3|}{36}]|^{81}_{1}=\frac{20}{243}-\frac{ln3}{12}+\frac{ln7}{36}$