Correct option: D.$\frac{{{x^{10}}}}{{2{{\left( {{x^5} + {x^3} + 1} \right)}^2}}} + c$
d
$\int {\frac{{2{x^{12}} + 5{x^9}}}{{{{\left[ {{x^5}\left( {1 + \frac{1}{{{x^2}}} + \frac{1}{{{x^5}}}} \right)} \right]}^3}}}} = \int {\frac{{2{x^{12}} + 5{x^9}}}{{{x^{15}}{{\left( {1 + \frac{1}{{{x^2}}} + \frac{1}{{{x^5}}}} \right)}^3}}}} dx$
Dividing numerator and denominator by $x^{15}$ we get,
$ = \int {\frac{{\frac{2}{{{x^3}}} + \frac{5}{{{x^6}}}}}{{{{\left( {1 + \frac{1}{{{x^2}}} + \frac{1}{{{x^5}}}} \right)}^3}}}} dx$
Put $\left(1+\frac{1}{x^{2}}+\frac{1}{x^{5}}\right)=t$
$=\int \frac{-d t}{t^{3}}$
$=\frac{-t^{-3+1}}{-3+1}+C=\frac{1}{2} \times \frac{1}{t^{2}}+C$
$=\frac{1}{2} \frac{1}{\left(1+\frac{1}{x^{2}}+\frac{1}{x^{5}}\right)^{2}}+C$
$=\frac{1}{2} \frac{x^{10}}{\left(x^{5}+x^{3}+1\right)^{2}}+C$