1. We need to evaluate $I=\int \frac{2x^{12}+5x^9}{\left(x^5+x^3+1\right)^3}\,dx.$

2. First, factor the numerator: $2x^{12}+5x^9=x^9(2x^3+5).$ So, $I=\int \frac{x^9(2x^3+5)}{\left(x^5+x^3+1\right)^3}\,dx.$

3. Notice the expression $x^5+x^3+1$ in the denominator. Let $u=x^5+x^3+1.$ Then $\frac{du}{dx}=5x^4+3x^2=x^2(5x^2+3),$ which does not directly match the numerator. So direct substitution is not convenient.

4. Now inspect the options. Since the denominator is $\left(x^5+x^3+1\right)^3$, a likely antiderivative is of the form $\frac{f(x)}{\left(x^5+x^3+1\right)^2}.$ We test option D: $F(x)=\frac{x^{10}}{2\left(x^5+x^3+1\right)^2}.$

5. Differentiate $F(x)$: $F(x)=\frac12 x^{10}(x^5+x^3+1)^{-2}.$ Using product rule, $F'(x)=\frac12\left[10x^9(x^5+x^3+1)^{-2}+x^{10}(-2)(x^5+x^3+1)^{-3}(5x^4+3x^2)\right].$ Simplify: $F'(x)=5x^9(x^5+x^3+1)^{-2}-x^{10}(5x^4+3x^2)(x^5+x^3+1)^{-3}.$

6. Take common denominator $\left(x^5+x^3+1\right)^3$: $F'(x)=\frac{5x^9(x^5+x^3+1)-x^{10}(5x^4+3x^2)}{(x^5+x^3+1)^3}.$

7. Expand the numerator: $5x^9(x^5+x^3+1)=5x^{14}+5x^{12}+5x^9,$ and $x^{10}(5x^4+3x^2)=5x^{14}+3x^{12}.$ So, $5x^{14}+5x^{12}+5x^9-(5x^{14}+3x^{12})=2x^{12}+5x^9.$

Thus, $F'(x)=\frac{2x^{12}+5x^9}{(x^5+x^3+1)^3},$ which exactly matches the integrand.

8. Therefore, $\int \frac{2x^{12}+5x^9}{(x^5+x^3+1)^3}\,dx=\frac{x^{10}}{2\left(x^5+x^3+1\right)^2}+C.$

9. Hence the correct option is D.