\documentclass[a4paper]{article}
\usepackage{amsmath}
\DeclareMathOperator{\arcosh}{arcosh}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \sqrt{a x^{2} - b x} \\
a
 & > & 0 \\
b
 & > & 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \sqrt{a x^{2} - b x} \, \mbox{d}x \\
& = & \int \sqrt{x (a x - b)} \, \mbox{d}x \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
x
 & = & \frac{b}{a \sin^{2} t} \\
\sin t
 & = & \sqrt{\frac{b}{a x}} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \sqrt{x (a x - b)} \, \mbox{d}x \\
& = & \int \sqrt{\frac{b}{a \sin^{2} t} \left ( a \frac{b}{a \sin^{2} t} - b \right )} \, \mbox{d} \left ( \frac{b}{a \sin^{2} t} \right ) \\
& = & \int \sqrt{\frac{b^{2}}{a \sin^{2} t} \left ( \frac{1}{\sin^{2} t} - 1 \right )} \frac{b}{a} \frac{- 2 \cos t}{\sin^{3} t} \, \mbox{d}t \\
& = & - 2 \int \frac{b}{\sqrt{a} \sin t} \sqrt{\frac{1}{\sin^{2} t} - \frac{\sin^{2} t}{\sin^{2} t}} \frac{b}{a} \frac{\cos t}{\sin^{3} t}
 \, \mbox{d}t \\
& = & - \frac{2 b^{2}}{a \sqrt{a}} \int \frac{\cos t}{\sin^{4} t} \sqrt{\frac{1 - \sin^{2} t}{\sin^{2} t}} \, \mbox{d}t \\
& = & - \frac{2 b^{2}}{a \sqrt{a}} \int \frac{\cos t}{\sin^{4} t} \sqrt{\frac{\cos^{2} t}{\sin^{2} t}} \, \mbox{d}t \\
& = & - \frac{2 b^{2}}{a \sqrt{a}} \int \frac{\cos t}{\sin^{4} t} \frac{\cos t}{\sin t} \, \mbox{d}t \\
& = & - \frac{2 b^{2}}{a \sqrt{a}} \int \frac{\cos^{2} t}{\sin^{5} t} \, \mbox{d}t \\
& = & - \frac{2 b^{2}}{a \sqrt{a}} \int \frac{1 - \sin^{2} t}{\sin^{5} t} \, \mbox{d}t \\
& = & \frac{2 b^{2}}{a \sqrt{a}} \int \left ( \frac{1}{\sin^{3} t} - \frac{1}{\sin^{5} t} \right ) \, \mbox{d}t \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{2 b^{2}}{a \sqrt{a}} \int \left ( \frac{1}{\sin^{3} t} - \frac{1}{\sin^{5} t} \right ) \, \mbox{d}t \\
& = & \frac{2 b^{2}}{a \sqrt{a}} \left ( - \frac{1}{2} \frac{\cos t}{\sin^{2} t}
 + \frac{1}{2} \ln \left | \, \frac{\sin t}{1 + \cos t} \, \right |
 + \frac{1}{4} \frac{\cos t}{\sin^{4} t}
 + \frac{3}{8} \frac{\cos t}{\sin^{2} t}
 - \frac{3}{8} \ln \left | \, \frac{\sin t}{1 + \cos t} \, \right | \right ) + c \\
& = & \frac{2 b^{2}}{a \sqrt{a}} \left ( \frac{1}{4} \frac{\cos t}{\sin^{4} t} - \frac{1}{8} \frac{\cos t}{\sin^{2} t}
 - \frac{1}{8} \ln \left | \, \frac{1 + \cos t}{\sin t} \, \right | \right ) + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{2 b^{2}}{a \sqrt{a}} \left ( \frac{1}{4} \frac{\cos t}{\sin^{4} t} - \frac{1}{8} \frac{\cos t}{\sin^{2} t}
 - \frac{1}{8} \ln \left | \, \frac{1 + \cos t}{\sin t} \, \right | \right ) + c \\
& = & \frac{2 b^{2}}{a \sqrt{a}} \left ( \frac{1}{4} \frac{\sqrt{1 - \sin^{2} t}}{\sin^{4} t} - \frac{1}{8} \frac{\sqrt{1 - \sin^{2} t}}{\sin^{2} t}
 - \frac{1}{8} \ln \left | \, \frac{1 + \sqrt{1 - \sin^{2} t}}{\sin t} \, \right | \right ) + c \\
& = & \frac{2 b^{2}}{a \sqrt{a}} \left (
 \frac{1}{4} \frac{\sqrt{1 - \left ( \sqrt{\frac{b}{a x}} \right )^{2}}}{\left ( \sqrt{\frac{b}{a x}} \right )^{4}}
 - \frac{1}{8} \frac{\sqrt{1 - \left ( \sqrt{\frac{b}{a x}} \right )^{2}}}{\left ( \sqrt{\frac{b}{a x}} \right )^{2}}
 - \frac{1}{8} \ln \left | \, \frac{1 + \sqrt{1 - \left ( \sqrt{\frac{b}{a x}} \right )^{2}}}{\sqrt{\frac{b}{a x}}} \, \right |
 \right ) + c \\
& = & \frac{2 b^{2}}{a \sqrt{a}} \left (
 \frac{1}{4} \frac{\sqrt{1 - \frac{b}{a x}}}{\left ( \frac{b}{a x} \right )^{2}}
 - \frac{1}{8} \frac{\sqrt{1 - \frac{b}{a x}}}{\frac{b}{a x}}
 - \frac{1}{8} \ln \left | \, \frac{1 + \sqrt{1 - \frac{b}{a x}}}{\sqrt{\frac{b}{a x}}} \, \right |
 \right ) + c \\
& = & \frac{2 b^{2}}{a \sqrt{a}} \left (
 \frac{1}{4} \frac{a x}{b} \sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1}
 - \frac{1}{8} \sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1}
 - \frac{1}{8} \ln \left | \, \sqrt{\frac{a x}{b}} + \sqrt{\frac{a x}{b} - 1} \, \right |
 \right ) + c \\
& = & \frac{b^{2}}{4 a \sqrt{a}} \left ( \left ( \frac{2 a x}{b} - 1 \right ) \sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1}
 - \ln \left | \, \sqrt{\frac{a x}{b}} + \sqrt{\frac{a x}{b} - 1} \, \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{b^{2}}{4 a \sqrt{a}} \left ( \left ( \frac{2 a x}{b} - 1 \right ) \sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1}
 - \ln \left | \, \sqrt{\frac{a x}{b}} + \sqrt{\frac{a x}{b} - 1} \, \right | \right ) + c \right ) \\
& = & \frac{b^{2}}{4 a \sqrt{a}} \left ( \left ( \frac{2 a}{b} - 0 \right ) \sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1}
 + \left ( \frac{2 a x}{b} - 1 \right ) \frac{1}{2} \frac{1}{\sqrt{\frac{a x}{b}}} \frac{a}{b} \sqrt{\frac{a x}{b} - 1} \right . \\
 & & \qquad \qquad \qquad \qquad
 + \, \left ( \frac{2 a x}{b} - 1 \right ) \sqrt{\frac{a x}{b}} \frac{1}{2} \frac{1}{\sqrt{\frac{a x}{b} - 1}} \frac{a}{b} \\
 & & \qquad \qquad \qquad \qquad \qquad
 \left . - \, \frac{1}{\sqrt{\frac{a x}{b}} + \sqrt{\frac{a x}{b} - 1}}
 \left ( \frac{1}{2} \frac{1}{\sqrt{\frac{a x}{b}}} \frac{a}{b} + \frac{1}{2} \frac{1}{\sqrt{\frac{a x}{b} - 1}} \frac{a}{b} \right ) \right ) + 0 \\
& = & \frac{b}{8 \sqrt{a}} \left ( 4 \sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1}
 + \left ( \frac{2 a x}{b} - 1 \right ) \frac{1}{\sqrt{\frac{a x}{b}}} \sqrt{\frac{a x}{b} - 1}
 + \left ( \frac{2 a x}{b} - 1 \right ) \sqrt{\frac{a x}{b}} \frac{1}{\sqrt{\frac{a x}{b} - 1}} \right . \\
 & & \qquad \qquad \qquad \qquad \qquad
 \left . - \, \frac{1}{\sqrt{\frac{a x}{b}} + \sqrt{\frac{a x}{b} - 1}}
 \left ( \frac{1}{\sqrt{\frac{a x}{b}}} + \frac{1}{\sqrt{\frac{a x}{b} - 1}} \right ) \right ) \\
& = & \frac{b}{8 \sqrt{a}}
 \left ( \left ( 4 + \left ( \frac{2 a x}{b} - 1 \right ) \frac{1}{\frac{a x}{b}} + \left ( \frac{2 a x}{b} - 1 \right ) \frac{1}{\frac{a x}{b} - 1} \right )
 \sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1} \right . \\
 & & \qquad \qquad \qquad \qquad \qquad
 \left . - \, \frac{1}{\sqrt{\frac{a x}{b}} + \sqrt{\frac{a x}{b} - 1}}
 \frac{\sqrt{\frac{a x}{b} - 1} + \sqrt{\frac{a x}{b}}}{\sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1}} \right ) \\
& = & \frac{b}{8 \sqrt{a}} \left ( \left ( 4 + 2 - \frac{1}{\frac{a x}{b}} + \frac{\frac{a x}{b}}{\frac{a x}{b} - 1} + 1 \right )
 \sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1} - \frac{1}{\sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1}} \right ) \\
& = & \frac{b}{8 \sqrt{a}} \left ( 7 - \frac{1}{\frac{a x}{b}} + \frac{\frac{a x}{b}}{\frac{a x}{b} - 1}
 - \frac{1}{\frac{a x}{b} \left ( \frac{a x}{b} - 1 \right )} \right ) \sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1} \\
& = & \frac{1}{8} \left ( 7 - \frac{1}{\frac{a x}{b}} + \frac{\frac{a x}{b}}{\frac{a x}{b} - 1}
 - \frac{1}{\frac{a x}{b} \left ( \frac{a x}{b} - 1 \right )} \right ) \sqrt{a x^{2} - b x} \\
& = & \frac{1}{8} \left ( \frac{7 \frac{a x}{b} \left ( \frac{a x}{b} - 1 \right )}{\frac{a x}{b} \left ( \frac{a x}{b} - 1 \right )}
 - \frac{\frac{a x}{b} - 1}{\frac{a x}{b} \left ( \frac{a x}{b} - 1 \right )}
 + \frac{\left ( \frac{a x}{b} \right )^{2}}{\frac{a x}{b} \left ( \frac{a x}{b} - 1 \right )}
 - \frac{1}{\frac{a x}{b} \left ( \frac{a x}{b} - 1 \right )} \right ) \sqrt{a x^{2} - b x} \\
& = & \frac{1}{8} \frac{7 \frac{a x}{b} \left ( \frac{a x}{b} - 1 \right ) - \frac{a x}{b} + 1
 + \left ( \frac{a x}{b} \right )^{2} - 1}{\frac{a x}{b} \left ( \frac{a x}{b} - 1 \right )} \sqrt{a x^{2} - b x} \\
& = & \frac{1}{8} \frac{7 \left ( \frac{a x}{b} \right )^{2} - 7 \frac{a x}{b} - \frac{a x}{b}
 + \left ( \frac{a x}{b} \right )^{2} - 1}{\frac{a x}{b} \left ( \frac{a x}{b} \right )} \sqrt{a x^{2} - b x} \\
& = & \frac{1}{8} \frac{8 \left ( \frac{a x}{b} \right )^{2} - 8 \frac{a x}{b}}{\frac{a x}{b} \left ( \frac{a x}{b} - 1 \right )} \sqrt{a x^{2} - b x} \\
& = & \frac{\left ( \frac{a x}{b} \right )^{2} - \frac{a x}{b}}{\frac{a x}{b} \left ( \frac{a x}{b} - 1 \right )} \sqrt{a x^{2} - b x} \\
& = & \sqrt{a x^{2} - b x} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\arcosh x
 & = & \ln \left ( x + \sqrt{x^{2} - 1} \, \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{b^{2}}{4 a \sqrt{a}} \left ( \left ( \frac{2 a x}{b} - 1 \right ) \sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1}
 - \ln \left | \, \sqrt{\frac{a x}{b}} + \sqrt{\frac{a x}{b} - 1} \, \right | \right ) + c \\
& = & \frac{b^{2}}{4 a \sqrt{a}} \left ( \left ( \frac{2 a x}{b} - 1 \right ) \sqrt{\frac{a x}{b}} \sqrt{\frac{a x}{b} - 1}
 - \arcosh \sqrt{\frac{a x}{b}} \, \right ) + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \pm \frac{b^{2}}{4 a \sqrt{a}} \left ( \pm \left ( \frac{2 a x}{b} - 1 \right ) \sqrt{\pm \frac{a x}{b}} \sqrt{\pm  \left ( \frac{a x}{b} - 1 \right )}
 - \ln \left | \, \sqrt{\pm \frac{a x}{b}} + \sqrt{\pm  \left ( \frac{a x}{b} - 1 \right )} \, \right | \right ) + c \\
& = & \pm \frac{b^{2}}{4 a \sqrt{a}} \left ( \pm \left ( \frac{2 a x}{b} - 1 \right ) \sqrt{\pm \frac{a x}{b}} \sqrt{\pm  \left ( \frac{a x}{b} - 1 \right )}
 - \arcosh \sqrt{\pm \frac{a x}{b}} \, \right ) + c \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \pm \frac{b^{2}}{4 a \sqrt{a}} \left ( \pm \left ( \frac{2 a x}{b} - 1 \right ) \sqrt{\pm \frac{a x}{b}} \sqrt{\pm  \left ( \frac{a x}{b} - 1 \right )}
 - \arcosh \sqrt{\pm \frac{a x}{b}} \, \right ) + c \\
& = & \pm \frac{b^{2}}{4 a \sqrt{a}} \left ( \pm \left ( \frac{2 a x}{b} - 1 \right ) \sqrt{\pm \frac{a x}{b}} \sqrt{\pm  \left ( \frac{a x}{b} - 1 \right )}
 - \ln \left | \, \sqrt{\pm \frac{a x}{b}} + \sqrt{\pm  \left ( \frac{a x}{b} - 1 \right )} \, \right | \right ) + c \\
\end{eqnarray*}
\end{document}
