$I=\int \dfrac{x^5 \sqrt{1+x^6}}{x^6}dx$ đặt $x^6 +1 =t^2 \Rightarrow x^5 dx = tdt$
$I=\int \dfrac{t.t}{t^2-1}dt=\int dt +\int \dfrac{1}{(t-1)(t+1)}dt=t+\dfrac{1}{2}\int \bigg (\dfrac{1}{t-1}-\dfrac{1}{t+1} \bigg )dt$
$=t+\dfrac{1}{2} \ln \bigg |\dfrac{t-1}{t+1} \bigg |+C=\sqrt{x^6 +1} +\dfrac{1}{2} \ln \bigg | \dfrac{\sqrt{x^6+1}-1}{\sqrt{x^6+1}+1} \bigg |+C$