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The intergral $\int_{}^{}\frac{2x^{12}+5x^{9}}{(x^{5}+x^{3}+1)^{3}} dx$ is equal to

where C is an arbitrary constant

A) $\frac{-x^{5}}{(x^{5}+x^{3}+1)^{2}} +C$

B) $\frac{x^{10}}{2(x^{5}+x^{3}+1)^{2}} +C$

C) $\frac{x^{5}}{2(x^{5}+x^{3}+1)^{2}} +C$

D) $\frac{-x^{10}}{2(x^{5}+x^{3}+1)^{2}} +C$

Option B

Let $I=\int_{}^{} \frac{2x^{2}+5x^{9}}{(x^{5}+x^{3}+1)^{3}}dx$

$=\int_{}^{} \frac{2x^{12}+5x^{9}}{x^{15}(1+x^{-2}+x^{-5})^{3}}dx$

$=\int_{}^{} \frac{2x^{-3}+5x^{-6}}{(1+x^{-2}+x^{-5})^{3}}dx$

Now put 1+x^-2+x^-5=t

$\Rightarrow (-2x^{-3}-5x^{-6})dx= dt$

$\Rightarrow (2x^{-3}+5x^{-6})dx=- dt$

$\therefore I= -\int_{}^{} \frac{dt}{t^{3}}=-\int_{}^{} t^{-3}dt$

$=-\frac{t^{-3+1}}{-3+1}+C= \frac{I}{2t^{2}}+C$

$\frac{x^{10}}{2(x^{5}+x^{3}+1)^{2}}+C$

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