The value of $\int e^{x} (\frac{x^{4} + 2}{{(1 + x^{2})}^{5 / 2}}) d x,$ is

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$\frac{e^{x} (1 - x + x^{2})}{{(1 + x^{2})}^{3 / 2}} + C$

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

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

$\frac{e^{x} (1 - x)}{{(1 + x^{2})}^{3 / 2}} + C$

answer is D.

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Detailed Solution

Let $I = \int e^{x} (\frac{x^{4} + 2}{{(1 + x^{2})}^{5 / 2}}) d x$
$\int e^{x} (\frac{1}{{(1 + x^{2})}^{1 / 2}} + \frac{1 - 2 x^{2}}{{(1 + x^{2})}^{5 / 2}}) d x \int e^{x} (\frac{1}{{(1 + x^{2})}^{1 / 2}} - \frac{x}{{(1 + x^{2})}^{3 / 2}} + \frac{x}{{(1 + x^{2})}^{3 / 2}} + \frac{1 - 2 x^{2}}{{(1 + x^{2})}^{5 / 2}}) d x = \frac{e^{x}}{{(1 + x^{2})}^{1 / 2}} + \frac{x e^{x}}{{(1 + x^{2})}^{3 / 2}} + C = \frac{e^{x} {1 + x^{2} + x}}{{(1 + x^{2})}^{3 / 2}} + C$