$\int \frac{\sqrt{5+x^{10}}}{x^{16}} d x$ is equal to:

$\frac{1}{75}\left(1+\frac{5}{x^{10}}\right)^{3 / 2}+c$

Let $I=\int \frac{\sqrt{5+x^{10}}}{x^{16}} dx$

Let $1+\frac{5}{x^{10}}=t \Rightarrow 5\left(\frac{-10}{x^{11}}\right) dx=dt$

$\frac{1}{x^{11}} dx=-\frac{1}{50} dt$

$\Rightarrow I=-\frac{1}{50} \int \sqrt{t} d t$

$=\frac{1}{50} \times \frac{2}{3} t^{3 / 2}=\frac{1}{75}\left(1+\frac{5}{x^{10}}\right)^{3 / 2}+c$

Hence (1) is the correct answer.