$\int \frac{x^{5 / 2}}{\sqrt{1+x^7}} d x$, is

$\frac{2}{7} \log \left|x^{7 / 2}+\sqrt{1+x^7}\right|+C$

Let $I=\int \frac{x^{5 / 2}}{\sqrt{1+x^7}} d x$

$\Rightarrow I=\frac{2}{7} \int \frac{1}{{\sqrt{1^2+\left(x^{7 / 2}\right)^2}}^{\frac{7}{2}} x^{\frac{5}{2}} d x}$

$\Rightarrow I=\frac{2}{7} \int \frac{1}{\sqrt{1^2+\left(x^{7 / 2}\right)^2}} d\left(x^{7 / 2}\right)=\frac{2}{7} \log \left|x^{7/2}+\sqrt{1+x^7}\right|+C$