Answer
$\dfrac {4}{15}\left( x^{3}+5\right) ^{\frac {5}{2}}+c$
Work Step by Step
$v=x^{3}+5\Rightarrow dv=3x^{2}dx$
$$\int 2x^{2}\left( x^{3}+5\right) ^{\frac {3}{2}}dx=\int \dfrac {2}{3}\times 3x^{2}\left( x^{3}+5\right) ^{\frac {3}{2}}dx=\dfrac {2}{3}\int v^{\frac {3}{2}}dv=\dfrac {2}{3}\times \dfrac {2}{5}v^{\frac {5}{2}}+c=\dfrac {4}{15}v^{\frac {5}{2}}+c=\dfrac {4}{15}\left( x^{3}+5\right) ^{\frac {5}{2}}+c$$
