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