[tex] \huge \mathcal \red{answer : }[/tex]

[tex] \sqrt{ {x}^{2} + 4 } + \frac{4}{ \sqrt{ {x}^{2} + 4 } } + c[/tex]

[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: [/tex]

[tex] \huge \mathcal \red{explanation:} \\ \\ ∫ \frac{ {x}^{3} }{( {x}^{2} + 4){}^{ \frac{3}{2} } }dx \\ \\ ∫ \frac{t - 4}{2t {}^{ \frac{3}{2} } }dt \\ \\ ( \frac{1}{2} )∫ \frac{t - 4}{ {t}^{ \frac{3}{2} } } dt \\ \\ ( \frac{1}{2} )∫ \frac{t - 4}{( {t}^{ \frac{3}{2} }) } dt \\ \\ ( \frac{1}{2} )∫2 \sqrt{t} + \frac{8}{ \sqrt{t} } \\ \\ = \sqrt{ {x}^{2} + 4} + \frac{4}{ \sqrt{ {x}^{2} + 4} } + c [/tex]

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