Explanation

(b)
Let $I = \int_{}^{}{\frac{2x^{12} + 5x^{9}}{\left( x^{5} + x^{3} + 1 \right)^{3}}\ dx}$
$= \int_{}^{}{\frac{\left( \frac{2}{x^{3}} + \frac{5}{x^{6}} \right)}{\left( 1 + \frac{1}{x^{2}} + \frac{1}{x^{5}} \right)^{3}}dx}$
Put $1 + \frac{1}{x^{2}} + \frac{1}{x^{5}} = t$
$\Rightarrow \left( \ - \frac{2}{x^{3}} - \frac{5}{x^{6}}\ \right)dx = dt$
Then, $I = - \int_{}^{}{\frac{dt}{t^{3}} = \frac{1}{2t^{2}} + c}$
$= \frac{1}{2\left( 1 + \frac{1}{x^{2}} + \frac{1}{x^{5}} \right)^{2}} + c$
$= \frac{x^{10}}{2\left( x^{5} + x^{3} + 1 \right)^{2}} + c$