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$\int \frac{2 x^{12} + 5 x^{9}}{{(x^{5} + x^{3} + 1)}^{3}} d x =$

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\frac{x^{10}}{2 {(x^{5} + x^{3} + 3)}^{2}} + c

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

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

\frac{x^{10}}{2 {(x^{4} + 2 x^{3} + 1)}^{2}} + c

answer is C.

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Detailed Solution

\int \frac{2 x^{12} + 5 x^{9}}{x^{15} {(1 + \frac{1}{x^{2}} + \frac{1}{x^{5}})}^{3}} d x

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

$[Let t = 1 + \frac{1}{x^{2}} + \frac{1}{x^{5}} \Rightarrow d t = (- 2 x^{- 3} - 5 x^{- 6}) d x]$

$- \int t^{- 3} d t = \frac{t^{- 2}}{2} + c = \frac{1}{2} {[1 + \frac{1}{x^{2}} + \frac{1}{x^{5}}]}^{- 2}$

$= \frac{1}{2} [\frac{x^{10}}{{(x^{5} + x^{3} + 1)}^{2}}] + c$ .

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