$\int (x^{7m}+x^{2m}+x^m){(2x^{6m}+7x^m+14)}^{\frac{1}{m}}dx=$

$I=\int (x^{7m}+x^{2m}+x^m){(2x^{6m}+7x^m+14)}^{\frac{1}{m}}dx$

$=\int (x^{7m-1}+x^{2m-1}+x^{m-1}).x. {\left[2.x^{6m}+7x^m+14\right]}^{\frac{1}{m}}dx$

$=\int {(x^{7m-1}+x^{2m-1}+x^{m-1}).\left[x^m\left(2.x^{6m}+7x^m+14\right)\right] }^{\frac{1}{m}}dx$

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

$Put 2x^7+7x^{2m}+14x^m=t$

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