\documentclass[a4paper]{article}
\usepackage{amsmath}
\DeclareMathOperator{\arcosh}{arcosh}
\begin{document}
\begin{eqnarray*}
f (x)
 & = & \sqrt{ax^{2} + bx + c} \\
a
 & > & 0 \\
D
 & = & b^{2} - 4ac \, \, \, > \, \, \, 0 \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f (x)
 & = & g (h (x)) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
f^{\prime} (x)
 & = & g^{\prime} (h (x)) h^{\prime} (x) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
p
 & = & \frac{- b + \sqrt{b^{2} - 4ac}}{2a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
q
 & = & \frac{- b - \sqrt{b^{2} - 4ac}}{2a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \sqrt{ax^{2} + bx + c} \, \mbox{d}x \\
& = & \int \sqrt{a (x - p) (x - q)} \, \mbox{d}x \\
& = & \int \sqrt{a \left ( x - \frac{- b + \sqrt{b^{2} - 4ac}}{2a} \right ) \left ( x - \frac{- b - \sqrt{b^{2} - 4ac}}{2a} \right )} \, \mbox{d}x \\
& = & \int \sqrt{a \left ( x + \frac{b}{2a} - \frac{\sqrt{b^{2} - 4ac}}{2a} \right )
 \left ( x + \frac{b}{2a} + \frac{\sqrt{b^{2} - 4ac}}{2a} \right )} \, \mbox{d}x \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
x
 & = & u - \frac{b}{2a} \\
u
 & = & x + \frac{b}{2a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \int \sqrt{a \left ( x + \frac{b}{2a} - \frac{\sqrt{b^{2} - 4ac}}{2a} \right )
 \left ( x + \frac{b}{2a} + \frac{\sqrt{b^{2} - 4ac}}{2a} \right )} \, \mbox{d}x \\
& = & \int \sqrt{a \left ( u - \frac{b}{2a} + \frac{b}{2a} - \frac{\sqrt{b^{2} - 4ac}}{2a} \right )
 \left ( u - \frac{b}{2a} + \frac{b}{2a} + \frac{\sqrt{b^{2} - 4ac}}{2a} \right )} \, \mbox{d} \left ( u - \frac{b}{2a} \right ) \\
& = & \int \sqrt{a \left ( u - \frac{\sqrt{b^{2} - 4ac}}{2a} \right ) \left ( u + \frac{\sqrt{b^{2} - 4ac}}{2a} \right )} \, \mbox{d}u \\
& = & \sqrt{a} \int \sqrt{u^{2} - \left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2}} \, \mbox{d}u \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \sqrt{a} \int \sqrt{u^{2} - \left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2}} \, \mbox{d}u \\
& = & \sqrt{a} \left ( \frac{1}{2} u \sqrt{u^{2} - \left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2}}
 - \frac{1}{2} \left ( \frac{\sqrt{b^{2} - 4ac}}{2a} \right )^{2} \arcosh \frac{2 a u}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & \sqrt{a} \left ( \frac{1}{2} u \sqrt{u^{2} - \frac{b^{2} - 4ac}{4a^{2}}}
 - \frac{1}{2} \frac{b^{2} - 4ac}{4a^{2}} \arcosh \frac{2 a u}{\sqrt{b^{2} - 4ac}} \right ) + C \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \sqrt{a} \left ( \frac{1}{2} u \sqrt{u^{2} - \frac{b^{2} - 4ac}{4a^{2}}}
 - \frac{1}{2} \frac{b^{2} - 4ac}{4a^{2}} \arcosh \frac{2 a u}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & \sqrt{a} \left ( \frac{1}{2} \left ( x + \frac{b}{2a} \right ) \sqrt{\left ( x + \frac{b}{2a} \right )^{2} - \frac{b^{2} - 4ac}{4a^{2}}}
 - \frac{1}{2} \frac{b^{2} - 4ac}{4a^{2}} \arcosh \frac{2 a \left ( x + \frac{b}{2a} \right )}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & \sqrt{a} \left ( \frac{1}{2} \left ( x + \frac{b}{2a} \right )
 \sqrt{x^{2} + \frac{b x}{a} + \frac{b^{2}}{4a^{2}} - \frac{b^{2}}{4a^{2}} + \frac{c}{a}}
 - \frac{1}{2} \frac{b^{2} - 4ac}{4a^{2}} \arcosh \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & \frac{1}{4a} \left ( (2 a x + b) \sqrt{ax^{2} + b x + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}} \arcosh \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \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}{4a} \left ( (2 a x + b) \sqrt{ax^{2} + b x + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}} \arcosh \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right ) + C \right ) \\
& = & \frac{1}{4a} \Biggl ( (2 a) \sqrt{ax^{2} + b x + c} + (2 a x + b) \frac{1}{2} \frac{1}{\sqrt{ax^{2} + b x + c}} (2ax + b) \\
 & & \qquad \qquad \qquad - \, \left . \frac{b^{2} - 4ac}{2 \sqrt{a}} \frac{1}{\sqrt{\left ( \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right )^{2} - 1}} \frac{2 a}{\sqrt{b^{2} - 4ac}} \right ) + 0 \\
& = & \frac{1}{4a} \left ( \frac{(2 a) (ax^{2} + b x + c) + \frac{1}{2} (2 a x + b)^{2}}{\sqrt{ax^{2} + b x + c}}
 - \sqrt{a} \sqrt{b^{2} - 4ac} \frac{1}{\sqrt{\frac{(2 a x + b)^{2}}{b^{2} - 4ac} - 1}} \right ) \\
& = & \frac{1}{4a} \left ( \frac{2 a^{2} x^{2} + 2 ab x + 2 ac + 2 a^{2} x^{2} + 2 ab x + \frac{1}{2} b^{2}}{\sqrt{ax^{2} + b x + c}}
 - \sqrt{a} (b^{2} - 4ac) \frac{1}{\sqrt{4 a^{2} x^{2} + 4 ab x + b^{2} - b^{2} + 4ac}} \right ) \\
& = & \frac{1}{4a} \left ( \frac{4 a^{2} x^{2} + 4 ab x + 2 ac + \frac{1}{2} b^{2}}{\sqrt{ax^{2} + b x + c}}
 - \sqrt{a} (b^{2} - 4ac) \frac{1}{\sqrt{4 a^{2} x^{2} + 4 ab x + 4ac}} \right ) \\
& = & \frac{1}{4a} \left ( \frac{4 a^{2} x^{2} + 4 ab x + 2 ac + \frac{1}{2} b^{2}}{\sqrt{ax^{2} + b x + c}}
 - \frac{1}{2} (b^{2} - 4ac) \frac{1}{\sqrt{a x^{2} + b x + c}} \right ) \\
& = & \frac{1}{4a} \frac{4 a^{2} x^{2} + 4 ab x + 2 ac + \frac{1}{2} b^{2} - \frac{1}{2} b^{2} + 2ac}{\sqrt{a x^{2} + b x + c}} \\
& = & \frac{1}{4a} \frac{4 a^{2} x^{2} + 4 ab x + 4 ac}{\sqrt{a x^{2} + b x + c}} \\
& = & \frac{a x^{2} + b x + c}{\sqrt{a x^{2} + b x + c}} \\
& = & \sqrt{a x^{2} + b x + c} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
\arcosh x
 & = & \ln \left ( x + \sqrt{x^{2} - 1} \, \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{4a} \left ( (2 a x + b) \sqrt{ax^{2} + b x + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}} \arcosh \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & \frac{1}{4a} \left ( (2 a x + b) \sqrt{ax^{2} + b x + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}} \ln \left ( \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} + \sqrt{\left ( \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right )^{2} - 1}
 \, \right ) \right ) + C \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{4a} \left ( (2 a x + b) \sqrt{ax^{2} + b x + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}} \ln \left | \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} + \sqrt{\left ( \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right )^{2} - 1}
 \, \right | \right ) + C \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{4a} \left ( (2 a x + b) \sqrt{ax^{2} + b x + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}} \arcosh \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right ) + C \\
& = & \frac{1}{4a} \left ( (2 a x + b) \sqrt{ax^{2} + b x + c}
 - \frac{D}{2 \sqrt{a}} \arcosh \frac{2 a x + b}{\sqrt{D}} \right ) + C \\
& = & \frac{1}{4a} \left ( (2 a x + b) \sqrt{ax^{2} + b x + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}} \ln \left | \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} + \sqrt{\left ( \frac{2 a x + b}{\sqrt{b^{2} - 4ac}} \right )^{2} - 1}
 \, \right | \right ) + C \\
& = & \frac{1}{4a} \left ( (2 a x + b) \sqrt{ax^{2} + b x + c}
 - \frac{D}{2 \sqrt{a}} \ln \left | \frac{2 a x + b}{\sqrt{D}} + \sqrt{\left ( \frac{2 a x + b}{\sqrt{D}} \right )^{2} - 1} \, \right | \right ) + C \\
\end{eqnarray*}
\end{document}
