Multiline equation inside a split environment

You can make use of mathtools' multlined environment:

\documentclass{article}
\usepackage{amsmath,mathtools}
\begin{document}
\[
\begin{split}
  x &= a + a \\
    &= b + b + b + b \\
    &= \!\begin{multlined}[t]
       c + c + c + c + c + c + c + c + c + c + c + c \\
         + c + c + c + c + c + c + c + c + c + c
     \end{multlined}
   \end{split}
\]
\end{document}

Note: the \! before the environment is to get correct spacing between = and c.

It's also possible to specify the total width of the two lines (from left margin at first line to right margin at last line) as an optional argument to multlined:

\documentclass{article}
\usepackage{amsmath,mathtools}
\begin{document}
\[
\begin{split}
  x &= a + a \\
    &= b + b + b + b \\
    &= \!\begin{multlined}[t][10cm]
       c + c + c + c + c + c + c + c + c + c + c + c \\
         + c + c + c + c + c + c + c + c + c + c
     \end{multlined}
   \end{split}
\]
\end{document}

If you want the two lines right aligned, you can set the mathtools key firstline-afterskip to 0pt, either globally or locally:

\documentclass[border=5pt,preview]{standalone}
\usepackage{amsmath,mathtools}
\begin{document}
\mathtoolsset{firstline-afterskip=0pt}
\[
\begin{split}
  x &= a + a \\
    &= b + b + b + b \\
    &= \!\begin{multlined}[t]
       c + c + c + c + c + c + c + c + c + c + c + c \\
         + c + c + c + c + c + c + c + c + c + c
     \end{multlined}
   \end{split}
\]
\end{document}

The standard and easiest method is to just add a line to the split and move right by, say, one quad. This is the method I'd prefer, not bothering about alignment.

Alternatively, use multlined from mathtools:

\documentclass{article}
\usepackage{amsmath,mathtools}
\begin{document}
\[
\begin{split}
x &= a + a \\
  &= b + b + b + b \\
  &= c + c + c + c + c + c + c + c + c + c + c + c \\
  &\qquad + c + c + c + c + c + c + c + c + c + c
\end{split}
\]
\[
\begin{split}
x &= a + a \\
  &= b + b + b + b \\
  &= \mathtoolsset{firstline-afterskip=0pt}
     \!\begin{multlined}[t] % <---- don't forget \!
     c + c + c + c + c + c + c + c + c + c + c + c \\
       + c + c + c + c + c + c + c + c + c + c
     \end{multlined}
\end{split}
\]
\end{document}

You could use a stack, for example. If the line heights were uniform, this would work:

\documentclass{article}
\usepackage{amsmath,stackengine}
\stackMath
\begin{document}
\[
\begin{split}
  x &= a + a \\
  &= b + b + b + b \\
  &= \stackengine{\baselineskip}{c + c + c + c + c + c + c + c + c + c + c + c}
  {{}+ c + c + c + c + c + c + c + c + c + c}{U}{r}{F}{F}{L}
\end{split}
\]
\end{document}

If the stacked rows were of non-standard heights, then this:

\documentclass{article}
\usepackage{amsmath,stackengine}
\stackMath
\begin{document}
\[
\begin{split}
  x &= a + a \\
  &= b + b + b + b \\
  &= \stackengine{5pt}{c + c + c + c + \dfrac{a}{b} + c + c + c + c + c + c + c}
  {{}+ c + \dfrac{c}{d} + c + c + c + c + c + c + c + c}{U}{r}{F}{F}{S}
\end{split}
\]
\end{document}

It is the 5th argument of \stackengine, given by {r} that defines the alignment of the stack. Other options are {l} and {c}.