<s>[INST] Evaluate the integral $\int \frac{1}{\sqrt[3]{(x+1)^2(x-2)^2}}\mathrm dx$ What substitution is useful for this integral?
$$\int \frac{1}{\sqrt[3]{(x+1)^2(x-2)^2}}\mathrm dx$$
Substitutions $u=x^{\frac{2}{3}},u=(x+1)^{\frac{2}{3}},u=(x-2)^{\frac{2}{3}}$ are not working. 
Can't find useful trigonometric substitution. 
 [/INST] As commented by Brevan Ellefsen, there is no elementary solution for $$I=\left(\frac{2}{3}\right)^{\frac{1}{3}}\int\frac{du}{\cos^{\frac{1}{3}}(u)}$$ but there is effectively a solution using hypergeometric function. It seems to be $$I=-\left(\frac{3}{2}\right)^{2/3} \cos ^{\frac{2}{3}}(u) \,
   _2F_1\left(\frac{1}{3},\frac{1}{2};\frac{4}{3};\cos ^2(u)\right)$$ where appears the Gaussian (sometimes called ordinary) hypergeometric function.
In terms of $x$, the antiderivative is then $$I=\int \frac{1}{\sqrt[3]{(x+1)^2(x-2)^2}}\, dx= \sqrt[3]{3(x-2)} \,\,\,
   _2F_1\left(\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{2-x}{3}\right)$$ Back to definitions (and getting rid of Pochhammer symbols), $$I=\sum_{n=0}^\infty \frac{3^{-4 n} \,\Gamma \left(\frac{1}{3}\right)\, \Gamma (3 n)}{n!\, \Gamma (n)\,
   \Gamma \left(n+\frac{4}{3}\right)} (2-x)^n$$
Edit
The coefficients in the last expression vary extremely fast ( the first  is $1$, the tenth  is $\approx 1.851\times 10^{-7}$, the hundredth is $\approx 1.025\times 10^{-51}$). For large values of $n$, they are $$\log(c_n)=-n \log (3)-\frac{4}{3} \log (n)+\log
   \left(\frac{\Gamma \left(\frac{1}{3}\right)}{2 \sqrt{3} \pi
   }\right)-\frac{4}{9 n}+O\left(\frac{1}{n^2}\right)$$
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