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

$\frac{\left(2 x^{7 m}+7 x^{2 m}+14 x^m\right)^{\frac{m+1}{m}}}{14(m+1)}+C$

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

Then,

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

Let $2 x^{7 m}+7 x^{2 m}+14 x^m=t$. Then,

$14 m\left(x^{7 m-1}+x^{2 m-1}+x^{m-1}\right) d x=d t$

∴ $I=\frac{1}{14 m} \int t^{1 / m} d t=\frac{1}{14 m} \frac{t^{\frac{1}{m}+1}}{\left(\frac{1}{m}+1\right)}+C$

$\Rightarrow I=\frac{1}{14(m+1)}\left(2 x^{7 m}+7 x^{2 m}+14 x^m\right)^{\frac{m+1}{m}}+C$