If $\int \frac{\sqrt{5+x^{10}}}{x^{16}} d x=a\left(1+\frac{5}{x^{10}}\right)^{3 / 2}+C$, $a=$

$-\frac{1}{75}$

We have,

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

$\Rightarrow I =\int \sqrt{\frac{5+x^{10}}{x^{10}}} \times \frac{1}{x^{11}} d x$

$\Rightarrow I =\frac{-1}{50} \int \sqrt{1+\frac{5}{x^{10}}} \times \frac{-50}{x^{11}} d x$

$\Rightarrow I=-\frac{1}{50} \int \sqrt{1+\frac{5}{x^{10}}} d\left(1+\frac{5}{x^{10}}\right)$

$\Rightarrow I=-\frac{1}{50} \times \frac{2}{3}\left(1+\frac{5}{x^{10}}\right)^{3 / 2}+C=-\frac{1}{75}\left(1+\frac{5}{x^{10}}\right)^{3 / 2}+C$

∴   $a=-\frac{1}{75}$