\documentclass[a4paper]{article}
\usepackage{amsmath}
\DeclareMathOperator{\arsinh}{arsinh}
\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} \\
& = & \frac{- b + i \sqrt{4ac - b^{2}}}{2a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
q
 & = & \frac{- b - \sqrt{b^{2} - 4ac}}{2a} \\
& = & \frac{- b - i \sqrt{4ac - b^{2}}}{2a} \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
i
 & = & \sqrt{- 1} \\
\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 + i \sqrt{4ac - b^{2}}}{2a} \right ) \left ( x - \frac{- b - i \sqrt{4ac - b^{2}}}{2a} \right )} \, \mbox{d}x \\
& = & \int \sqrt{a \left ( x + \frac{b}{2a} - i \frac{\sqrt{4ac - b^{2}}}{2a} \right )
 \left ( x + \frac{b}{2a} + i \frac{\sqrt{4ac - b^{2}}}{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} - i \frac{\sqrt{4ac - b^{2}}}{2a} \right )
 \left ( x + \frac{b}{2a} + i \frac{\sqrt{4ac - b^{2}}}{2a} \right )} \, \mbox{d}x \\
& = & \int \sqrt{a \left ( u - \frac{b}{2a} + \frac{b}{2a} - i \frac{\sqrt{4ac - b^{2}}}{2a} \right )
 \left ( u - \frac{b}{2a} + \frac{b}{2a} + i \frac{\sqrt{4ac - b^{2}}}{2a} \right )} \, \mbox{d} \left ( u - \frac{b}{2a} \right ) \\
& = & \int \sqrt{a \left ( u - i \frac{\sqrt{4ac - b^{2}}}{2a} \right ) \left ( u + i \frac{\sqrt{4ac - b^{2}}}{2a} \right )} \, \mbox{d}u \\
& = & \sqrt{a} \int \sqrt{u^{2} + \left ( \frac{\sqrt{4ac - b^{2}}}{2a} \right )^{2}} \, \mbox{d}u \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \sqrt{a} \int \sqrt{u^{2} + \left ( \frac{\sqrt{4ac - b^{2}}}{2a} \right )^{2}} \, \mbox{d}u \\
& = & \sqrt{a} \left ( \frac{1}{2} u \sqrt{\left ( \frac{\sqrt{4ac - b^{2}}}{2a} \right )^{2} + u^{2}}
 + \frac{1}{2} \left ( \frac{\sqrt{4ac - b^{2}}}{2a} \right )^{2} \arsinh \frac{2au}{\sqrt{4ac - b^{2}}} \right ) + C \\
& = & \frac{1}{2} \sqrt{a} \left ( u \sqrt{\frac{4ac - b^{2}}{4a^{2}} + u^{2}}
 + \frac{4ac - b^{2}}{4a^{2}} \arsinh \frac{2au}{\sqrt{4ac - b^{2}}} \right ) + C \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{2} \sqrt{a} \left ( u \sqrt{\frac{4ac - b^{2}}{4a^{2}} + u^{2}}
 + \frac{4ac - b^{2}}{4a^{2}} \arsinh \frac{2au}{\sqrt{4ac - b^{2}}} \right ) + C \\
& = & \frac{1}{2} \sqrt{a} \left ( \left ( x + \frac{b}{2a} \right ) \sqrt{\frac{4ac - b^{2}}{4a^{2}} + \left ( x + \frac{b}{2a} \right )^{2}}
 + \frac{4ac - b^{2}}{4a^{2}} \arsinh \frac{2a \left ( x + \frac{b}{2a} \right )}{\sqrt{4ac - b^{2}}} \right ) + C \\
& = & \frac{1}{8 a \sqrt{a}} \left ( (2ax + b) \sqrt{4ac - b^{2} + (2ax + b)^{2}}
 + (4ac - b^{2}) \arsinh \frac{2ax + b}{\sqrt{4ac - b^{2}}} \right ) + C \\
& = & \frac{1}{8 a \sqrt{a}} \left ( (2ax + b) \sqrt{4ac - b^{2} + 4a^{2}x^{2} + 4abx + b^{2}}
 + (4ac - b^{2}) \arsinh \frac{2ax + b}{\sqrt{4ac - b^{2}}} \right ) + C \\
& = & \frac{1}{8 a \sqrt{a}} \left ( (2ax + b) \sqrt{4a (ax^{2} + bx + c)}
 + (4ac - b^{2}) \arsinh \frac{2ax + b}{\sqrt{4ac - b^{2}}} \right ) + C \\
& = & \frac{1}{4a} \left ( (2ax + b) \sqrt{ax^{2} + bx + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}} \arsinh \frac{2ax + b}{\sqrt{4ac - b^{2}}} \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}} \arsinh \frac{2 a x + b}{\sqrt{4ac - b^{2}}} \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{4ac - b^{2}}} \right )^{2} + 1}}
 \frac{2 a}{\sqrt{4ac - b^{2}}} \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{4ac - b^{2}} \frac{1}{\sqrt{\frac{(2 a x + b)^{2}}{4ac - b^{2}} + 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} (4ac - b^{2}) \frac{1}{\sqrt{4 a^{2} x^{2} + 4 ab x + b^{2} + 4ac - b^{2}}} \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} (4ac - b^{2}) \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} (4ac - b^{2}) \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} + 2ac - \frac{1}{2} b^{2}}{\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*}
\arsinh x
 & = & \ln \left ( x + \sqrt{x^{2} + 1} \, \right ) \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{4a} \left ( (2ax + b) \sqrt{ax^{2} + bx + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}} \arsinh \frac{2ax + b}{\sqrt{4ac - b^{2}}} \right ) + C \\
& = & \frac{1}{4a} \left ( (2ax + b) \sqrt{ax^{2} + bx + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}}
 \ln \left ( \frac{2 a x + b}{\sqrt{4ac - b^{2}}} + \sqrt{\left ( \frac{2 a x + b}{\sqrt{4ac - b^{2}}} \right )^{2} + 1} \, \right ) \right ) + C \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{4a} \left ( (2ax + b) \sqrt{ax^{2} + bx + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}}
 \ln \left | \, \frac{2 a x + b}{\sqrt{4ac - b^{2}}} + \sqrt{\left ( \frac{2 a x + b}{\sqrt{4ac - b^{2}}} \right )^{2} + 1} \, \right | \right ) + C \\
\end{eqnarray*}
\newpage
\begin{eqnarray*}
F (x)
 & = & \frac{1}{4a} \left ( (2ax + b) \sqrt{ax^{2} + bx + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}} \arsinh \frac{2ax + b}{\sqrt{4ac - b^{2}}} \right ) + C \\
& = & \frac{1}{4a} \left ( (2ax + b) \sqrt{ax^{2} + bx + c}
 - \frac{D}{2 \sqrt{a}} \arsinh \frac{2ax + b}{\sqrt{- D}} \right ) + C \\
& = & \frac{1}{4a} \left ( (2ax + b) \sqrt{ax^{2} + bx + c}
 - \frac{b^{2} - 4ac}{2 \sqrt{a}}
 \ln \left | \, \frac{2 a x + b}{\sqrt{4ac - b^{2}}} + \sqrt{\left ( \frac{2 a x + b}{\sqrt{4ac - b^{2}}} \right )^{2} + 1} \, \right | \right ) + C \\
& = & \frac{1}{4a} \left ( (2ax + b) \sqrt{ax^{2} + bx + 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}
