Integrand size = 39, antiderivative size = 119 \[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\frac {\sqrt [3]{-1+x^3}}{-1+x^2}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^3}}{-2+2 x^2+\sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1-x^2+\sqrt [3]{-1+x^3}\right )-\frac {1}{6} \log \left (1-2 x^2+x^4+\left (-1+x^2\right ) \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
[Out] (x^3-1)^(1/3)/(x^2-1)-1/3*arctan(3^(1/2)*(x^3-1)^(1/3)/(-2+2*x^2+(x^3- 1)^(1/3)))*3^(1/2)+1/3*ln(1-x^2+(x^3-1)^(1/3))-1/6*ln(1-2*x^2+x^4+(x^2 -1)*(x^3-1)^(1/3)+(x^3-1)^(2/3))
\[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx \]
[In] Int[((2 + x + x^2)*(-1 + x^3)^(1/3))/(x*(-1 + x^2)^2*(-3 - 2*x + x^2 + x^3)),x]
[Out] 1/(2*(-1 + x^3)^(2/3)) + x^2/(2*(-1 + x^3)^(2/3)) - (-1 + x^3)^(1/3)/3 - (-1 + x^3)^(1/3)/(2*(1 + x^3)) + (x*(-1 + x^3)^(1/3))/(2*(1 + x^3)) - (x^2*(-1 + x^3)^(1/3))/(2*(1 + x^3)) - (2*ArcTan[(1 - 2*(-1 + x^3)^ (1/3))/Sqrt[3]])/(3*Sqrt[3]) - (x*(1 - x^3)^(2/3)*Hypergeometric2F1[1/ 3, 2/3, 4/3, x^3])/(6*(-1 + x^3)^(2/3)) + (x*(1 - x^3)^(2/3)*Hypergeom etric2F1[1/3, 5/3, 4/3, x^3])/(3*(-1 + x^3)^(2/3)) + (x^4*(1 - x^3)^(2 /3)*Hypergeometric2F1[4/3, 5/3, 7/3, x^3])/(6*(-1 + x^3)^(2/3)) - Log[ x]/3 + Log[1 + (-1 + x^3)^(1/3)]/3 + (2*Defer[Int][(-1 + x^3)^(1/3)/(- 3 - 2*x + x^2 + x^3), x])/3 + Defer[Int][(x*(-1 + x^3)^(1/3))/(-3 - 2* x + x^2 + x^3), x]/3 + Defer[Int][(x^2*(-1 + x^3)^(1/3))/(-3 - 2*x + x ^2 + x^3), x]/3
Rubi steps \begin{align*} \text {integral}= \int \left (-\frac {\sqrt [3]{-1+x^3}}{3 (-1+x)^2}+\frac {\sqrt [3]{-1+x^3}}{12 (-1+x)}-\frac {2 \sqrt [3]{-1+x^3}}{3 x}+\frac {\sqrt [3]{-1+x^3}}{2 (1+x)^2}+\frac {\sqrt [3]{-1+x^3}}{4 (1+x)}+\frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{3 \left (-3-2 x+x^2+x^3\right )}\right ) \, dx \\ = \frac {1}{12} \int \frac {\sqrt [3]{-1+x^3}}{-1+x} \, dx+\frac {1}{4} \int \frac {\sqrt [3]{-1+x^3}}{1+x} \, dx-\frac {1}{3} \int \frac {\sqrt [3]{-1+x^3}}{(-1+x)^2} \, dx+\frac {1}{3} \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3} \, dx+\frac {1}{2} \int \frac {\sqrt [3]{-1+x^3}}{(1+x)^2} \, dx-\frac {2}{3} \int \frac {\sqrt [3]{-1+x^3}}{x} \, dx \\ = \frac {1}{12} \int \left (\frac {1}{\left (-1+x^3\right )^{2/3}}+\frac {x}{\left (-1+x^3\right )^{2/3}}+\frac {x^2}{\left (-1+x^3\right )^{2/3}}\right ) \, dx-\frac {2}{9} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x} \, dx,x,x^3\right )+\frac {1}{4} \int \left (\frac {\sqrt [3]{-1+x^3}}{1+x^3}-\frac {x \sqrt [3]{-1+x^3}}{1+x^3}+\frac {x^2 \sqrt [3]{-1+x^3}}{1+x^3}\right ) \, dx-\frac {1}{3} \int \left (\frac {1}{\left (-1+x^3\right )^{5/3}}+\frac {2 x}{\left (-1+x^3\right )^{5/3}}+\frac {3 x^2}{\left (-1+x^3\right )^{5/3}}+\frac {2 x^3}{\left (-1+x^3\right )^{5/3}}+\frac {x^4}{\left (-1+x^3\right )^{5/3}}\right ) \, dx+\frac {1}{3} \int \left (\frac {2 \sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3}+\frac {x \sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3}+\frac {x^2 \sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3}\right ) \, dx+\frac {1}{2} \int \left (\frac {\sqrt [3]{-1+x^3}}{\left (1+x^3\right )^2}-\frac {2 x \sqrt [3]{-1+x^3}}{\left (1+x^3\right )^2}+\frac {3 x^2 \sqrt [3]{-1+x^3}}{\left (1+x^3\right )^2}-\frac {2 x^3 \sqrt [3]{-1+x^3}}{\left (1+x^3\right )^2}+\frac {x^4 \sqrt [3]{-1+x^3}}{\left (1+x^3\right )^2}\right ) \, dx \\ = -\frac {2}{3} \sqrt [3]{-1+x^3}+\frac {1}{12} \int \frac {1}{\left (-1+x^3\right )^{2/3}} \, dx+\frac {1}{12} \int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx+\frac {1}{12} \int \frac {x^2}{\left (-1+x^3\right )^{2/3}} \, dx+\frac {2}{9} \text {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right )+\frac {1}{4} \int \frac {\sqrt [3]{-1+x^3}}{1+x^3} \, dx-\frac {1}{4} \int \frac {x \sqrt [3]{-1+x^3}}{1+x^3} \, dx+\frac {1}{4} \int \frac {x^2 \sqrt [3]{-1+x^3}}{1+x^3} \, dx-\frac {1}{3} \int \frac {1}{\left (-1+x^3\right )^{5/3}} \, dx-\frac {1}{3} \int \frac {x^4}{\left (-1+x^3\right )^{5/3}} \, dx+\frac {1}{3} \int \frac {x \sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3} \, dx+\frac {1}{3} \int \frac {x^2 \sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3} \, dx+\frac {1}{2} \int \frac {\sqrt [3]{-1+x^3}}{\left (1+x^3\right )^2} \, dx+\frac {1}{2} \int \frac {x^4 \sqrt [3]{-1+x^3}}{\left (1+x^3\right )^2} \, dx-\frac {2}{3} \int \frac {x}{\left (-1+x^3\right )^{5/3}} \, dx-\frac {2}{3} \int \frac {x^3}{\left (-1+x^3\right )^{5/3}} \, dx+\frac {2}{3} \int \frac {\sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3} \, dx+\frac {3}{2} \int \frac {x^2 \sqrt [3]{-1+x^3}}{\left (1+x^3\right )^2} \, dx-\int \frac {x^2}{\left (-1+x^3\right )^{5/3}} \, dx-\int \frac {x \sqrt [3]{-1+x^3}}{\left (1+x^3\right )^2} \, dx-\int \frac {x^3 \sqrt [3]{-1+x^3}}{\left (1+x^3\right )^2} \, dx \\ = \frac {1}{2 \left (-1+x^3\right )^{2/3}}+\frac {x^2}{2 \left (-1+x^3\right )^{2/3}}-\frac {7}{12} \sqrt [3]{-1+x^3}+\frac {x \sqrt [3]{-1+x^3}}{2 \left (1+x^3\right )}-\frac {x^2 \sqrt [3]{-1+x^3}}{2 \left (1+x^3\right )}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{12 \sqrt {3}}-\frac {\log (x)}{3}-\frac {1}{24} \log \left (x-\sqrt [3]{-1+x^3}\right )+\frac {1}{12} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{1+x} \, dx,x,x^3\right )-\frac {1}{6} \int \frac {2-x^3}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx+\frac {1}{6} \int \frac {x \left (-2+3 x^3\right )}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx-\frac {1}{4} \int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx-\frac {1}{3} \int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx+\frac {1}{3} \int \frac {x}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx+\frac {1}{3} \int \frac {x \sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3} \, dx+\frac {1}{3} \int \frac {x^2 \sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3} \, dx-\frac {1}{3} \int \frac {-1+2 x^3}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {1}{2} \int \frac {x}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx+\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{(1+x)^2} \, dx,x,x^3\right )+\frac {2}{3} \int \frac {\sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3} \, dx+\frac {9}{4} \text {Subst}\left (\int \frac {x}{\left (1-2 x^3\right ) \left (4+x^3\right )} \, dx,x,\frac {1-x}{\sqrt [3]{-1+x^3}}\right )+\frac {\left (1-x^3\right )^{2/3}}{12 \left (-1+x^3\right )^{2/3}} \int \frac {1}{\left (1-x^3\right )^{2/3}} \, dx+\frac {\left (1-x^3\right )^{2/3}}{3 \left (-1+x^3\right )^{2/3}} \int \frac {1}{\left (1-x^3\right )^{5/3}} \, dx+\frac {2 \left (1-x^3\right )^{2/3}}{3 \left (-1+x^3\right )^{2/3}} \int \frac {x^3}{\left (1-x^3\right )^{5/3}} \, dx \\ = \frac {1}{2 \left (-1+x^3\right )^{2/3}}+\frac {x^2}{2 \left (-1+x^3\right )^{2/3}}-\frac {1}{3} \sqrt [3]{-1+x^3}-\frac {\sqrt [3]{-1+x^3}}{2 \left (1+x^3\right )}+\frac {x \sqrt [3]{-1+x^3}}{2 \left (1+x^3\right )}-\frac {x^2 \sqrt [3]{-1+x^3}}{2 \left (1+x^3\right )}+\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {5 \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{2} x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{6\ 2^{2/3} \sqrt {3}}+\frac {x \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {4}{3};x^3\right )}{12 \left (-1+x^3\right )^{2/3}}+\frac {x \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {1}{3},\frac {5}{3};\frac {4}{3};x^3\right )}{3 \left (-1+x^3\right )^{2/3}}+\frac {x^4 \left (1-x^3\right )^{2/3} \, _2F_1\left (\frac {4}{3},\frac {5}{3};\frac {7}{3};x^3\right )}{6 \left (-1+x^3\right )^{2/3}}-\frac {\log (x)}{3}+\frac {5 \log \left (1+x^3\right )}{36\ 2^{2/3}}+\frac {1}{4} \log \left (x-\sqrt [3]{-1+x^3}\right )-\frac {5 \log \left (\sqrt [3]{2} x-\sqrt [3]{-1+x^3}\right )}{12\ 2^{2/3}}+\frac {1}{3} \log \left (1+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \int \frac {1}{\left (-1+x^3\right )^{2/3}} \, dx+\frac {1}{6} \int \left (\frac {3 x}{\left (-1+x^3\right )^{2/3}}-\frac {5 x}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )}\right ) \, dx+\frac {1}{4} \text {Subst}\left (\int \frac {x}{4+x^3} \, dx,x,\frac {1-x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{3} \int \frac {x \sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3} \, dx+\frac {1}{3} \int \frac {x^2 \sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3} \, dx-\frac {1}{2} \int \frac {1}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx+\frac {1}{2} \text {Subst}\left (\int \frac {x}{1-2 x^3} \, dx,x,\frac {1-x}{\sqrt [3]{-1+x^3}}\right )-\frac {2}{3} \int \frac {1}{\left (-1+x^3\right )^{2/3}} \, dx+\frac {2}{3} \int \frac {\sqrt [3]{-1+x^3}}{-3-2 x+x^2+x^3} \, dx-\frac {2}{3} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right )+\int \frac {1}{\left (-1+x^3\right )^{2/3} \left (1+x^3\right )} \, dx \\ = \text {Too large to display} \\ \end{align*}
Time = 5.41 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\frac {\sqrt [3]{-1+x^3}}{-1+x^2}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^3}}{-2+2 x^2+\sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1-x^2+\sqrt [3]{-1+x^3}\right )-\frac {1}{6} \log \left (1-2 x^2+x^4+\left (-1+x^2\right ) \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
[In] Integrate[((2 + x + x^2)*(-1 + x^3)^(1/3))/(x*(-1 + x^2)^2*(-3 - 2*x + x^2 + x^3)),x]
[Out] (-1 + x^3)^(1/3)/(-1 + x^2) - ArcTan[(Sqrt[3]*(-1 + x^3)^(1/3))/(-2 + 2*x^2 + (-1 + x^3)^(1/3))]/Sqrt[3] + Log[1 - x^2 + (-1 + x^3)^(1/3)]/3 - Log[1 - 2*x^2 + x^4 + (-1 + x^2)*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3 )]/6
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.46 (sec) , antiderivative size = 1329, normalized size of antiderivative = 11.17
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1329\) |
| risch | \(\text {Expression too large to display}\) | \(1795\) |
[In] int((x^2+x+2)*(x^3-1)^(1/3)/x/(x^2-1)^2/(x^3+x^2-2*x-3),x,method=_RETU RNVERBOSE)
[Out] (x^3-1)^(1/3)/(x^2-1)+3/2*RootOf(81*_Z^2+18*_Z+4)*ln(-(333182903538021 92*x+33719421708885792*(x^3-1)^(1/3)*x^3+33719421708885792*x^2*(x^3-1) ^(1/3)+14747439992666544*x^4-29494879985333088*x^3-33719421708885792*( x^3-1)^(1/3)-10924029624197440*x^2+33719421708885792*(x^3-1)^(2/3)+454 57696558242270*RootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(1/3)*x^3+4545769655824 2270*RootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(2/3)*x+45457696558242270*RootOf( 81*_Z^2+18*_Z+4)*(x^3-1)^(1/3)*x^2-45457696558242270*RootOf(81*_Z^2+18 *_Z+4)*(x^3-1)^(1/3)*x+14747439992666544*x^5-19642219640243838*RootOf( 81*_Z^2+18*_Z+4)*x^5-19642219640243838*RootOf(81*_Z^2+18*_Z+4)*x^4+392 84439280487676*RootOf(81*_Z^2+18*_Z+4)*x^3+102579264381097584*RootOf(8 1*_Z^2+18*_Z+4)*x^2+43652605460366070*RootOf(81*_Z^2+18*_Z+4)*x+284301 8479155015*RootOf(81*_Z^2+18*_Z+4)^2*x^5-8122909940442900*RootOf(81*_Z ^2+18*_Z+4)^2+43652605460366070*RootOf(81*_Z^2+18*_Z+4)+33719421708885 792*x*(x^3-1)^(2/3)-33719421708885792*(x^3-1)^(1/3)*x+2843018479155015 *RootOf(81*_Z^2+18*_Z+4)^2*x^4-5686036958310030*RootOf(81*_Z^2+18*_Z+4 )^2*x^3-16651965377907945*RootOf(81*_Z^2+18*_Z+4)^2*x^2-81229099404429 00*RootOf(81*_Z^2+18*_Z+4)^2*x+45457696558242270*RootOf(81*_Z^2+18*_Z+ 4)*(x^3-1)^(2/3)-45457696558242270*RootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(1/ 3)+33318290353802192)/(x^3+x^2-2*x-3)/x^2)-1/3*ln(-(23216580007526132* x+23617711362609732*(x^3-1)^(1/3)*x^3+23617711362609732*x^2*(x^3-1)^(1 /3)+19252773664777768*x^4-38505547329555536*x^3-23617711362609732*(x^3 -1)^(1/3)-34541740986807172*x^2+23617711362609732*(x^3-1)^(2/3)-454576 96558242270*RootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(1/3)*x^3-4545769655824227 0*RootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(2/3)*x-45457696558242270*RootOf(81* _Z^2+18*_Z+4)*(x^3-1)^(1/3)*x^2+45457696558242270*RootOf(81*_Z^2+18*_Z +4)*(x^3-1)^(1/3)*x+19252773664777768*x^5+20905783408757178*RootOf(81* _Z^2+18*_Z+4)*x^5+20905783408757178*RootOf(81*_Z^2+18*_Z+4)*x^4-418115 66817514356*RootOf(81*_Z^2+18*_Z+4)*x^3-109980137882390004*RootOf(81*_ Z^2+18*_Z+4)*x^2-47262787656118470*RootOf(81*_Z^2+18*_Z+4)*x+284301847 9155015*RootOf(81*_Z^2+18*_Z+4)^2*x^5-8122909940442900*RootOf(81*_Z^2+ 18*_Z+4)^2-47262787656118470*RootOf(81*_Z^2+18*_Z+4)+23617711362609732 *x*(x^3-1)^(2/3)-23617711362609732*(x^3-1)^(1/3)*x+2843018479155015*Ro otOf(81*_Z^2+18*_Z+4)^2*x^4-5686036958310030*RootOf(81*_Z^2+18*_Z+4)^2 *x^3-16651965377907945*RootOf(81*_Z^2+18*_Z+4)^2*x^2-8122909940442900* RootOf(81*_Z^2+18*_Z+4)^2*x-45457696558242270*RootOf(81*_Z^2+18*_Z+4)* (x^3-1)^(2/3)+45457696558242270*RootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(1/3)+ 23216580007526132)/(x^3+x^2-2*x-3)/x^2)-3/2*ln(-(23216580007526132*x+2 3617711362609732*(x^3-1)^(1/3)*x^3+23617711362609732*x^2*(x^3-1)^(1/3) +19252773664777768*x^4-38505547329555536*x^3-23617711362609732*(x^3-1) ^(1/3)-34541740986807172*x^2+23617711362609732*(x^3-1)^(2/3)-454576965 58242270*RootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(1/3)*x^3-45457696558242270*R ootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(2/3)*x-45457696558242270*RootOf(81*_Z^ 2+18*_Z+4)*(x^3-1)^(1/3)*x^2+45457696558242270*RootOf(81*_Z^2+18*_Z+4) *(x^3-1)^(1/3)*x+19252773664777768*x^5+20905783408757178*RootOf(81*_Z^ 2+18*_Z+4)*x^5+20905783408757178*RootOf(81*_Z^2+18*_Z+4)*x^4-418115668 17514356*RootOf(81*_Z^2+18*_Z+4)*x^3-109980137882390004*RootOf(81*_Z^2 +18*_Z+4)*x^2-47262787656118470*RootOf(81*_Z^2+18*_Z+4)*x+284301847915 5015*RootOf(81*_Z^2+18*_Z+4)^2*x^5-8122909940442900*RootOf(81*_Z^2+18* _Z+4)^2-47262787656118470*RootOf(81*_Z^2+18*_Z+4)+23617711362609732*x* (x^3-1)^(2/3)-23617711362609732*(x^3-1)^(1/3)*x+2843018479155015*RootO f(81*_Z^2+18*_Z+4)^2*x^4-5686036958310030*RootOf(81*_Z^2+18*_Z+4)^2*x^ 3-16651965377907945*RootOf(81*_Z^2+18*_Z+4)^2*x^2-8122909940442900*Roo tOf(81*_Z^2+18*_Z+4)^2*x-45457696558242270*RootOf(81*_Z^2+18*_Z+4)*(x^ 3-1)^(2/3)+45457696558242270*RootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(1/3)+232 16580007526132)/(x^3+x^2-2*x-3)/x^2)*RootOf(81*_Z^2+18*_Z+4)
Exception generated. \[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\text {Exception raised: TypeError} \]
[In] integrate((x^2+x+2)*(x^3-1)^(1/3)/x/(x^2-1)^2/(x^3+x^2-2*x-3),x, algor ithm="fricas")
[Out] Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (residue poly has multiple non-li near factors)
Timed out. \[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\text {Timed out} \]
[In] integrate((x**2+x+2)*(x**3-1)**(1/3)/x/(x**2-1)**2/(x**3+x**2-2*x-3),x )
[Out] Timed out
\[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + x + 2\right )}}{{\left (x^{3} + x^{2} - 2 \, x - 3\right )} {\left (x^{2} - 1\right )}^{2} x} \,d x } \]
[In] integrate((x^2+x+2)*(x^3-1)^(1/3)/x/(x^2-1)^2/(x^3+x^2-2*x-3),x, algor ithm="maxima")
[Out] integrate((x^3 - 1)^(1/3)*(x^2 + x + 2)/((x^3 + x^2 - 2*x - 3)*(x^2 - 1)^2*x), x)
\[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + x + 2\right )}}{{\left (x^{3} + x^{2} - 2 \, x - 3\right )} {\left (x^{2} - 1\right )}^{2} x} \,d x } \]
[In] integrate((x^2+x+2)*(x^3-1)^(1/3)/x/(x^2-1)^2/(x^3+x^2-2*x-3),x, algor ithm="giac")
[Out] integrate((x^3 - 1)^(1/3)*(x^2 + x + 2)/((x^3 + x^2 - 2*x - 3)*(x^2 - 1)^2*x), x)
Timed out. \[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\int -\frac {{\left (x^3-1\right )}^{1/3}\,\left (x^2+x+2\right )}{x\,{\left (x^2-1\right )}^2\,\left (-x^3-x^2+2\,x+3\right )} \,d x \]
[In] int(-((x^3 - 1)^(1/3)*(x + x^2 + 2))/(x*(x^2 - 1)^2*(2*x - x^2 - x^3 + 3)),x)
[Out] int(-((x^3 - 1)^(1/3)*(x + x^2 + 2))/(x*(x^2 - 1)^2*(2*x - x^2 - x^3 + 3)), x)