How to typeset text along the curve of a mathematical function with TikZ

Question

I want to label a family of curves representing mathematical functions by writing at some place of the curve the formula directly on the trace. More specifically, my family of curves is the very simple x -> x²+c where c will take various values between -4 and 4. As this family of functions will be difficult to label the usual way (with formula next to the curve), I want to superpose the label directly on the curve (thus interrupting the trace - with a little white space to help the reading).

I tried with TikZ, and failed. The following code is OK with decorations like "bumps", but not with "text along path".

\documentclass[11pt,a4paper]{article}
\usepackage{tikz}
\usetikzlibrary{arrows, arrows.meta, decorations.text, decorations.pathmorphing}

\begin{document}

    \begin{center}
        \begin{tikzpicture}[>=Stealth ]
        \def\xmin{-4}
        \def\xmax{4}
        \def\ymin{-6}
        \def\ymax{6}
        \draw[thin, opacity=0.2] (\xmin, \ymin) grid (\xmax, \ymax);
        \draw[very thick, ->] (\xmin,0) -- (\xmax,0) node[below left] {$x$} ;
        \draw[very thick, ->] (0,\ymin) -- (0,\ymax) node[below left] {$y$} ;

        \clip (\xmin, \ymin) rectangle (\xmax, \ymax) ;
        \draw[very thick, dashed, domain=-3:3,samples=200] plot (\x, {(\x)^2} ) ;
        \draw[very thick, domain=\xmin:\xmax,samples=200, red, decorate, 
                decoration={bumps} % OK
                %decoration={text along path, text={\small test}} % Not OK
                ]
            plot (\x, {(\x)^2+3} ) ;
        \draw[decorate, decoration={text along path, text={Some long text along a ridiculously long curve that could yet be longer}} ] (-4,3) to[in=180,out=0] (0,0) to[in=180,out=0] (4,3) ; % OK too
        \end{tikzpicture}
    \end{center}

\end{document}

This code does not compile, in an odd way : the compilation never ends ! I use LuaLaTeX with the latest TexLive (on Ubuntu 18.04, but this is pointless).

Additionally, I found something near to my needs here : Pgfplots: plot text along function. But this approach uses PGFplots syntax which I do not master - and don't have enough time to begin mastering now.

Does someone have any idea ? Thanks in advance !

Update

Thanks to Schrödinger's Cat, who seems to be quite alive, I'm close to what I have in mind. But I have major difficulties to surround the label on each curve to mask the curve, in order to improve its readbility. Here a code showing where I am :

\documentclass[11pt,a4paper]{article}
\usepackage{tikz}
\usetikzlibrary{arrows, arrows.meta, decorations.text, decorations.pathmorphing} % decoration.markings

\begin{document}
        \begin{tikzpicture}[scale=.8,>=Stealth]
            \def \xmin {-4}
            \def \xmax {4}
            \def \ymin {-6}
            \def \ymax {6}
            \draw[thin, opacity=0.2] (\xmin, \ymin) grid (\xmax, \ymax);

            \draw[very thick, ->] (\xmin,0) -- (\xmax,0) node[below left] {$x$} ;
            \draw[very thick, ->] (0,\ymin) -- (0,\ymax) node[below left] {$y$} ;

            \clip (\xmin, \ymin) rectangle (\xmax, \ymax) ;
            \draw[very thick, dashed, domain=-3:3,samples=200] plot (\x, {(\x)^2} ) ;
            \foreach \c [evaluate=\c as \pc using {100-(\c+6)*9}, evaluate=\c as \ic using {abs(\xmax-\c)+.5}] in {-6,-4,-2,2,4} {
                \draw[very thick, domain=\xmin:\xmax,samples=200, blue!\pc!red,
                %                  text effects={path from text, characters={text along path, fill=gray!50}},
                postaction={decorate,
                    decoration={text along path,
                        raise=-.5ex,
%                        fill=green     % Let (Lua)LaTeX hang
                        text={|\scriptsize| $x {\mapsto} {x^2}+(\c)$},
                        text color=blue!\pc!red,
                        text align={align=right, right indent=\ic cm}
                    },
                    text effects={path from text, characters={text along path, fill=gray!50}},
                } ]
                plot (\x, {(\x)^2+\c} )  ;
            } ;

            \draw (0,0) node[shift={(-.3cm,-.3cm)}, fill=white, circle, inner sep=-.5pt] {$O$} ;
            \coordinate (I) at (1,0);  \draw (I) node {\small $|$} node[yshift=-.3cm, fill=white, inner sep=.5pt] {$1$} ;
            \coordinate (J) at (0,1);  \draw (J) node {$-$} node[xshift=-.2cm, fill=white, inner sep=.5pt] {$1$} ;
        \end{tikzpicture}
\end{document}

Answer 1

As is written in section 50.6 Text Decorations of the pgfmanual, you need to wrap text format commands in the corresponding delimiters. The default delimiters are | on both sides. Using this, and adding text align={align=center} to avoid that the text gets clipped away, one gets

\documentclass[11pt,a4paper]{article}
\usepackage{tikz}
\usetikzlibrary{arrows.meta, decorations.text}

\begin{document}

    \begin{center}
        \begin{tikzpicture}[>=Stealth ]
        \def\xmin{-4}
        \def\xmax{4}
        \def\ymin{-6}
        \def\ymax{6}
        \draw[thin, opacity=0.2] (\xmin, \ymin) grid (\xmax, \ymax);
        \draw[very thick, ->] (\xmin,0) -- (\xmax,0) node[below left] {$x$} ;
        \draw[very thick, ->] (0,\ymin) -- (0,\ymax) node[below left] {$y$} ;

        \clip (\xmin, \ymin) rectangle (\xmax, \ymax) ;
        \draw[very thick, dashed, domain=-3:3,samples=200] plot (\x, {(\x)^2} ) ;
        \draw[very thick, domain=\xmin:\xmax,samples=200, red, 
        postaction={decorate, 
                decoration={text along path, text={|\small|Some long text along a ridiculously long curve that could yet be longer},
                text align={align=center}}} % Now OK
                ]
            plot (\x, {(\x)^2+3} ) ;
        \draw[decorate, decoration={text along path, text={Some long text along a ridiculously long curve that could yet be longer}} ] (-4,3) to[in=180,out=0] (0,0) to[in=180,out=0] (4,3) ; % OK too
        \end{tikzpicture}
    \end{center}

\end{document}

Notice that I used postaction=decorate to have both the parabola and the text, and that you could also use parabola bend instead of a plot with 200 samples.