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

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

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

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

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

answer is A.

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Detailed Solution

$\int \frac{2 x^{12} + 5 x^{9}}{x^{15} {[1 + \frac{1}{x^{2}} + \frac{1}{x^{5}}]}^{3}} d x = \int \frac{\frac{2}{x^{3}} + \frac{5}{x^{6}}}{{[1 + x^{- 2} + x^{- 5}]}^{3}} d x$

Put $1 + x^{- 2} + x^{- 5} = t$

Diff

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

$= \int \frac{1}{t^{3}} (- d t) = \frac{1}{2 t^{2}} + c = \frac{x^{10}}{2 {[x^{5} + x^{3} + 1]}^{2}} + c$

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