You can use [`FourierCoefficient`](http://reference.wolfram.com/mathematica/ref/FourierCoefficient.html) to pre-calculate the Fourier coefficients to arbitrary degree and then use the result very effectively.

There may be some issues with zero-th degree, therefore I excluded this using `Piecewise`. Here's the main code block:

    f[x_] := Piecewise[{
        {-x^3 - 2 x, -2 < x < 0},
        {-1 + x, 0 <= x <= 2}},
        0
    ];
        Module[{x, fp},
        (* Set parameters so that the integration runs
           from -2 to 2 *)
        fp = {0, Pi/2};
        fc = Piecewise[{
            {FourierCoefficient[f[x], x, 0, FourierParameters -> fp], # == 0},
            {ComplexExpand@FourierCoefficient[f[x], x, #, FourierParameters -> fp], True}
        }] &;
        fc = Evaluate /@ fc;
    ];

The output (`fc`) of this is something very unpleasant to look at; however, there's nothing Fourier-related left there, all the hard math has been done already, and only a bunch of elementary functions remain. `fc` is now a function of one argument that gives you the n-th Fourier coefficient of `f` in no time.

    (* Calculate the first 2001 Fourier coefficients *)
    AbsoluteTiming[Table[fc[n], {n, -1000, 1000}]] // First

>     0.777059 seconds

To convert this back to the function, you have to do the partial sum with your hands, for example here are the second, fourth and eighieth partial sums:

    myPartialSums = Table[
        (* 1/2 and Pi/2 compensate the custom FourierParameters, see
           documentation of FourierSeries/FourierParameters
           under "more info" *)
        1/2 Re[Sum[fc[k] Exp[Pi/2 I k t], {k, -n, n}]],
        {n, {2, 4, 80}}
    ];

>     A very large output has been generated,
    but we're luckily not interested in it
    anyway but would rather plot it

    Plot[
        {f[t]}~Join~myPartialSums,
        {t, -10, 10},
        PlotRange -> All, 
        Evaluated -> True, 
        PlotStyle -> {Thick, Automatic, Automatic, Automatic}
    ]

>![enter image description here][1]


  [1]: https://i.sstatic.net/Yc6Mz.png