For any natural number m,

$\int \left(x^{7m}+x^{2m}+x^m\right){\left(2x^{6m}+7x^m+14\right)}^{1/m}dx$, where x>0 equals

Let $I=\int \left(x^{7m}+x^{2m}+x^{m\mathrm{\prime }}\right){\left(2x^{6m}+7x^m+14\right)}^{1/m}dx$

Then,

$I=\int \left(x^{7m-1}+x^{2m-1}+x^{m-1}\right){\left(2x^{7m}+7x^{2m}+14x^m\right)}^{1/m}dx$

Let $2x^{7m}+7x^{2m}+14x^m=t$. Then,

$\begin{array}{l}14m\left(x^{7m-1}+x^{2m-1}+x^{m-1}\right)dx=dt\\ \therefore l=\frac{1}{14m}\int t^{1/m}dt=\frac{1}{14m}\frac{t^{\frac{1}{m}+1}}{\left(\frac{1}{m}+1\right)}+C\\ \Rightarrow I=\frac{1}{14(m+1)}{\left(2x^{7m}+7x^{2m}+14x^m\right)}^{\frac{m+1}{m}}+C\end{array}$