let x=t6⇒dx=6t5dt=∫1+t33t26t5dt=6∫t71+3t3+3t6+t9dt=6∫t7+3t10+3t13+t16dt=6x4/38+3x11/611+x7/314+x17/617+C

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$\int {(1 + \sqrt{x})}^{3} x^{\frac{1}{3}} d x =____$

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$5 [\frac{x^{\frac{4}{3}}}{8} + 3 \frac{x^{\frac{11}{6}}}{11} + 3 \frac{x^{\frac{7}{3}}}{14} + \frac{x^{\frac{17}{6}}}{17}] + C$

$4 [\frac{x^{\frac{4}{3}}}{8} + 3 \frac{x^{\frac{11}{6}}}{11} + 3 \frac{x^{\frac{7}{3}}}{14} + \frac{x^{\frac{17}{6}}}{17}] + C$

$6 [\frac{x^{\frac{4}{3}}}{8} + 3 \frac{x^{\frac{11}{6}}}{11} + 3 \frac{x^{\frac{7}{3}}}{14} + \frac{x^{\frac{17}{6}}}{17}] + C$

$8 [\frac{x^{\frac{4}{3}}}{8} + 3 \frac{x^{\frac{11}{6}}}{11} + 3 \frac{x^{\frac{7}{3}}}{14} + \frac{x^{\frac{17}{6}}}{17}] + C$

answer is C.

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Detailed Solution

$\begin{array}{l} let x = t^{6} \Rightarrow d x = 6 t^{5} d t \\ = \int {(1 + t^{3})}^{3} t^{2} 6 t^{5} d t \\ = 6 \int t^{7} (1 + 3 t^{3} + 3 t^{6} + t^{9}) d t \\ = 6 \int t^{7} + 3 t^{10} + 3 t^{13} + t^{16} d t \\ = 6 [\frac{x^{4 / 3}}{8} + 3 \frac{x^{11 / 6}}{11} + \frac{x^{7 / 3}}{14} + \frac{x^{17 / 6}}{17}] + C \end{array}$