\documentclass[a4paper]{article}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{\frac{3}{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
x
 & = & a \sin^{3} t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\sin t
 & = & \sqrt[3]{\frac{x}{a}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
t
 & = & \arcsin \sqrt[3]{\frac{x}{a}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{\frac{3}{2}} \, \mbox{d}x \\
& = & \int \left ( 1 - \left ( \sqrt[3]{\frac{x}{a}} \, \right )^{2} \right )^{\frac{3}{2}} \, \mbox{d}x \\
& = & \int (1 - (\sin t)^{2})^{\frac{3}{2}} \, \mbox{d}(a \sin^{3} t) \\
& = & \int (1 - \sin^{2} t)^{\frac{3}{2}} 3a \sin^{2} t \cos t \, \mbox{d}t \\
& = & 3a \int (\cos^{2} t)^{\frac{3}{2}} (1 - \cos^{2} t) \cos t \, \mbox{d}t \\
& = & 3a \int \cos^{3} t (\cos t - \cos^{3} t) \, \mbox{d}t \\
& = & 3a \int (\cos^{4} t - \cos^{6} t) \, \mbox{d}t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & 3a \int (\cos^{4} t - \cos^{6} t) \, \mbox{d}t \\
& = & 3a \Biggl ( \left ( \frac{3}{8} t + \frac{3}{8} \cos t \sin t + \frac{1}{4} \cos^{3} t \sin t \right ) \\
 & & \qquad \qquad \qquad
 - \, \left ( \frac{5}{16} t + \frac{5}{16} \cos t \sin t + \frac{5}{24} \cos^{3} t \sin t + \frac{1}{6} \cos^{5} t \sin t \right ) \Biggr ) + c \\
& = & 3a \left ( \frac{3}{8} t + \frac{3}{8} \cos t \sin t + \frac{1}{4} \cos^{3} t \sin t
 - \frac{5}{16} t - \frac{5}{16} \cos t \sin t - \frac{5}{24} \cos^{3} t \sin t - \frac{1}{6} \cos^{5} t \sin t \right ) + c \\
& = & 3a \left ( \frac{1}{16} t + \frac{1}{16} \cos t \sin t + \frac{1}{24} \cos^{3} t \sin t - \frac{1}{6} \cos^{5} t \sin t \right ) + c \\
& = & \frac{1}{16} a (3 t + 3 \cos t \sin t + 2 \cos^{3} t \sin t - 8 \cos^{5} t \sin t) + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{16} a (3 t + 3 \cos t \sin t + 2 \cos^{3} t \sin t - 8 \cos^{5} t \sin t) + c \\
& = & \frac{1}{16} a \left ( 3 t + 3 \sqrt{1 - \sin^{2} t} \sin t + 2 \left ( \sqrt{1 - \sin^{2} t} \, \right )^{3} \sin t
 - 8 \left ( \sqrt{1 - \sin^{2} t} \, \right )^{5} \sin t \right ) + c \\
& = & \frac{1}{16} a \left ( 3 \arcsin \sqrt[3]{\frac{x}{a}}
 + 3 \sqrt{1 - \left ( \sqrt[3]{\frac{x}{a}} \, \right )^{2}} \sqrt[3]{\frac{x}{a}}
 + 2 \left ( \sqrt{1 - \left ( \sqrt[3]{\frac{x}{a}} \, \right )^{2}} \, \right )^{3} \sqrt[3]{\frac{x}{a}} \right . \\
 & & \qquad \qquad \qquad \qquad
 \left . - \, 8 \left ( \sqrt{1 - \left ( \sqrt[3]{\frac{x}{a}} \, \right )^{2}} \, \right )^{5} \sqrt[3]{\frac{x}{a}} \right ) + c \\
& = & \frac{1}{16} a \left ( 3 \arcsin \sqrt[3]{\frac{x}{a}}
 + \sqrt[3]{\frac{x}{a}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( 3 + 2 \left ( \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}} \, \right )^{2}
 - 8 \left ( \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}} \, \right )^{4} \right ) \right ) + c \\
& = & \frac{1}{16} a \left ( 3 \arcsin \sqrt[3]{\frac{x}{a}}
 + \sqrt[3]{\frac{x}{a}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( 3 + 2 \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )
 - 8 \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{2} \right ) \right ) + c \\
& = & \frac{1}{16} a \left ( 3 \arcsin \sqrt[3]{\frac{x}{a}}
 + \sqrt[3]{\frac{x}{a}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( 3 + 2 - 2 \left ( \frac{x}{a} \right )^{\frac{2}{3}}
 - 8 \left ( 1 - 2 \left ( \frac{x}{a} \right )^{\frac{2}{3}} + \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \right ) \right ) + c \\
& = & \frac{1}{16} a \left ( 3 \arcsin \sqrt[3]{\frac{x}{a}}
 + \sqrt[3]{\frac{x}{a}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( 3 + 2 - 2 \left ( \frac{x}{a} \right )^{\frac{2}{3}}
 - 8 + 16 \left ( \frac{x}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \right ) + c \\
& = & \frac{1}{16} a \left ( 3 \arcsin \sqrt[3]{\frac{x}{a}}
 + \sqrt[3]{\frac{x}{a}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( - 3 + 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \right ) + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & \frac{\mbox{d}(F (x))}{\mbox{d}x} \\
& = & \frac{\mbox{d}}{\mbox{d}x} \left ( \frac{1}{16} a \left ( 3 \arcsin \sqrt[3]{\frac{x}{a}}
 + \sqrt[3]{\frac{x}{a}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( - 3 + 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \right ) + c \right ) \\
& = & \frac{\mbox{d}}{\mbox{d}x} \left ( \frac{1}{16} a \left ( 3 \arcsin \left ( \frac{x}{a} \right )^{\frac{1}{3}}
 + \left ( \frac{x}{a} \right )^{\frac{1}{3}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( - 3 + 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \right ) + c \right ) \\
& = & \frac{1}{16} a \left (
 3 \frac{1}{\sqrt{1 - \left ( \left ( \frac{x}{a} \right )^{\frac{1}{3}} \right )^{2}}} \frac{1}{3} \left ( \frac{x}{a} \right )^{- \frac{2}{3}} \frac{1}{a}
 + \frac{1}{3} \left ( \frac{x}{a} \right )^{- \frac{2}{3}} \frac{1}{a} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( - 3 + 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \right . \\
 & & \qquad \qquad \qquad + \, \left ( \frac{x}{a} \right )^{\frac{1}{3}} \frac{1}{2} \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}}
 \left ( 0 - \frac{2}{3} \left ( \frac{x}{a} \right )^{- \frac{1}{3}} \frac{1}{a} \right )
 \left ( - 3 + 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \\
 & & \qquad \qquad \qquad \qquad \qquad \left . + \, \left ( \frac{x}{a} \right )^{\frac{1}{3}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( 0 + 14 \cdot \frac{2}{3} \left ( \frac{x}{a} \right )^{- \frac{1}{3}} \frac{1}{a}
 - 8 \cdot \frac{4}{3} \left ( \frac{x}{a} \right )^{\frac{1}{3}} \frac{1}{a} \right )
 \right ) + 0 \\
& = & \frac{1}{48} \left (
 3 \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}} \left ( \frac{x}{a} \right )^{- \frac{2}{3}}
 + \left ( \frac{x}{a} \right )^{- \frac{2}{3}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( - 3 + 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \right . \\
 & & \qquad \qquad \qquad - \, \left ( \frac{x}{a} \right )^{\frac{1}{3}} \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}}
 \left ( \frac{x}{a} \right )^{- \frac{1}{3}}
 \left ( - 3 + 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \\
 & & \qquad \qquad \qquad \qquad \qquad \left . + \, \left ( \frac{x}{a} \right )^{\frac{1}{3}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( 28 \left ( \frac{x}{a} \right )^{- \frac{1}{3}} - 32 \left ( \frac{x}{a} \right )^{\frac{1}{3}} \right ) \right ) \\
& = & \frac{1}{48} \left (
 3 \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}} \left ( \frac{x}{a} \right )^{- \frac{2}{3}}
 + \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( - 3 \left ( \frac{x}{a} \right )^{- \frac{2}{3}} + 14 - 8 \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right ) \right . \\
 & & \qquad \qquad \qquad \left . - \, \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}}
 \left ( - 3 + 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right )
 + \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( 28 - 32 \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right ) \right ) \\
& = & \frac{1}{48} \left (
 \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}}
 \left ( 3 \left ( \frac{x}{a} \right )^{- \frac{2}{3}}
 + 3 - 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} + 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \right . \\
 & & \qquad \qquad \qquad \qquad \qquad \left . + \, \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}
 \left ( - 3 \left ( \frac{x}{a} \right )^{- \frac{2}{3}} + 42 - 40 \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right ) \right ) \\
& = & \frac{1}{48} \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}} \left (
 3 \left ( \frac{x}{a} \right )^{- \frac{2}{3}}
 + 3 - 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} + 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right . \\
 & & \qquad \qquad \qquad \qquad \qquad \left . + \, \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )
 \left ( - 3 \left ( \frac{x}{a} \right )^{- \frac{2}{3}} + 42 - 40 \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right ) \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & \frac{1}{48} \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}} \left (
 3 \left ( \frac{x}{a} \right )^{- \frac{2}{3}}
 + 3 - 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} + 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right . \\
 & & \qquad \qquad \qquad \qquad \qquad \left . + \, \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )
 \left ( - 3 \left ( \frac{x}{a} \right )^{- \frac{2}{3}} + 42 - 40 \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right ) \right ) \\
& = & \frac{1}{48} \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}} \left (
 3 \left ( \frac{x}{a} \right )^{- \frac{2}{3}}
 + 3 - 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} + 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right . \\
 & & \qquad \qquad \qquad \qquad \qquad \left .
 - \, 3 \left ( \frac{x}{a} \right )^{- \frac{2}{3}} + 42 - 40 \left ( \frac{x}{a} \right )^{\frac{2}{3}}
 + 3 - 42 \left ( \frac{x}{a} \right )^{\frac{2}{3}} + 40 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \\
& = & \frac{1}{48} \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}} \left (
 48 - 96 \left ( \frac{x}{a} \right )^{\frac{2}{3}} + 48 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \\
& = & \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}}
 \left ( 1 - 2 \left ( \frac{x}{a} \right )^{\frac{2}{3}} + \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \\
& = & \frac{1}{\sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}}} \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{2} \\
& = & \left ( 1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}} \right )^{\frac{3}{2}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{16} a \Biggl ( 3 \arcsin \sqrt[3]{\frac{x}{a}}
 + \sqrt[3]{\frac{x}{a}} \sqrt{1 - \left ( \frac{x}{a} \right )^{\frac{2}{3}}} \\
 & & \qquad \qquad \cdot \left ( - 3 + 14 \left ( \frac{x}{a} \right )^{\frac{2}{3}} - 8 \left ( \frac{x}{a} \right )^{\frac{4}{3}} \right ) \Biggr ) + c \\
\end{eqnarray*}
\end{document}
