Cases with long math equation

with \mfrac from nccmath and multlined from the mathtools package:

\documentclass{article}
\usepackage{mathtools, nccmath}

%---------------- show page layout. don't use in a real document!
\usepackage{showframe}
\renewcommand\ShowFrameLinethickness{0.15pt}
\renewcommand*\ShowFrameColor{\color{red}}
%---------------------------------------------------------------%

\begin{document}

\[
V_{ijk} = \begin{cases} 
  \mfrac{y_u+y_l}{2} - \mfrac{x_u+x_l}{2},  &\text{if }x_u < y_l    \\[3ex]
    \begin{multlined} 
    \mfrac{1}{C} \bigg[\mfrac{x_u^3-y_l^3}{6}+\mfrac{y_l^2+x_u^2}{2}x_l
      + (x_u-y_l)\mfrac{x_l^2}{2}\\[-2ex]
      + \mfrac{y_u^2-x_u^2}{2}(x_u-x_l) - (y_u-x_u)\mfrac{x_u^2-x_l^2}{2} \biggr]
    \end{multlined},
    & \text{if }x_u\in [y_l,y_u ]
    \\[5ex]
    \mfrac{1}{x_u-x_l}\biggl[\mfrac{(y_u+y_l)^2}{6} - \mfrac{y_u+y_l}{2} x_l
    + \mfrac{x_l^2}{2}\biggr],& \text{otherwise}
  \end{cases}
 \]
\end{document}

I'm pretty sure it is a duplicate. But it is also easy to solve using aligned

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{align*}
  V_{ijk}& = \begin{cases} \dfrac{y_u+y_l}2 - \dfrac{x_u+x_l}2,&
    \text{if }x_u < y_l
    \\[15pt]
    \begin{aligned}[b]
      &\dfrac1C \bigg[\dfrac{x_u^3-y_l^3}{6}+\dfrac{y_l^2+x_u^2}{2}x_l
      + (x_u-y_l)\dfrac{x_l^2}{2}
      \\
      &+ \dfrac{y_u^2-x_u^2}{2}(x_u-x_l) -
      (y_u-x_u)\dfrac{x_u^2-x_l^2}{2} \bigg]
    \end{aligned},
    & \text{if }x_u\in\left[y_l,y_u\right]
    \\[15pt]
    \dfrac1{x_u-x_l}\bigg[\dfrac{(y_u+y_l)^2}6 - \dfrac{y_u+y_l}2 x_l
    + \dfrac{x_l^2}2\bigg],& \text{otherwise}
  \end{cases}
 \end{align*}
\end{document}

Here's a solution that employs a dcases* environment instead of the cases environment. The d in dcases denotes "display style"; this means, among other things, that one can write \frac instead of \dfrac in order to get "large" fraction terms.

Note that I've removed unnecessary \left and \right directives.

\documentclass{article}
\usepackage{mathtools} % for 'dcases*' environment

\begin{document}
\[
V_{ijk} = 
\begin{dcases*} 
\frac{y_u+y_l}{2} - \frac{x_u+x_l}{2}\,,
   &if $x_u < y_l$\\[1ex]
\begin{aligned}[b]
&\frac{1}{C} \biggl[
   \frac{x_u^3-y_l^3}{6}+\frac{y_l^2+x_u^2}{2}x_l
   + (x_u-y_l)\frac{x_l^2}{2}\\
&\quad + \frac{y_u^2-x_u^2}{2}(x_u-x_l) 
   - (y_u-x_u)\frac{x_u^2-x_l^2}{2} \,\biggr]\,,
\end{aligned}
   &if $x_u\in[y_l,y_u]$ \\[1ex]
\frac{1}{x_u-x_l} \biggl[ 
   \frac{(y_u+y_l)^2}{6} 
   - \frac{y_u+y_l}{2} x_l 
   + \frac{x_l^2}{2} \,\biggr]\,,
   & otherwise
\end{dcases*}
\]
\end{document}