Integrand size = 48, antiderivative size = 156 \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\frac {\sqrt [3]{-2+x^2+2 x^3} \left (12-12 x^2+54 x^3+3 x^4-27 x^5-38 x^6\right )}{4 x^4 \left (-2+x^2+x^3\right )}-\frac {7 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-2+x^2+2 x^3}}\right )}{\sqrt {3}}-\frac {7}{3} \log \left (-x+\sqrt [3]{-2+x^2+2 x^3}\right )+\frac {7}{6} \log \left (x^2+x \sqrt [3]{-2+x^2+2 x^3}+\left (-2+x^2+2 x^3\right )^{2/3}\right ) \]
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\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 \sqrt [3]{-2+x^2+2 x^3}}{5 (-1+x)^2}-\frac {17 \sqrt [3]{-2+x^2+2 x^3}}{5 (-1+x)}+\frac {6 \sqrt [3]{-2+x^2+2 x^3}}{x^5}-\frac {\sqrt [3]{-2+x^2+2 x^3}}{x^3}+\frac {9 \sqrt [3]{-2+x^2+2 x^3}}{x^2}-\frac {8 (3+x) \sqrt [3]{-2+x^2+2 x^3}}{5 \left (2+2 x+x^2\right )^2}+\frac {(4+17 x) \sqrt [3]{-2+x^2+2 x^3}}{5 \left (2+2 x+x^2\right )}\right ) \, dx \\ & = \frac {1}{5} \int \frac {(4+17 x) \sqrt [3]{-2+x^2+2 x^3}}{2+2 x+x^2} \, dx+\frac {2}{5} \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{(-1+x)^2} \, dx-\frac {8}{5} \int \frac {(3+x) \sqrt [3]{-2+x^2+2 x^3}}{\left (2+2 x+x^2\right )^2} \, dx-\frac {17}{5} \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{-1+x} \, dx+6 \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{x^5} \, dx+9 \int \frac {\sqrt [3]{-2+x^2+2 x^3}}{x^2} \, dx-\int \frac {\sqrt [3]{-2+x^2+2 x^3}}{x^3} \, dx \\ & = \frac {1}{5} \int \left (\frac {(17+13 i) \sqrt [3]{-2+x^2+2 x^3}}{(2-2 i)+2 x}+\frac {(17-13 i) \sqrt [3]{-2+x^2+2 x^3}}{(2+2 i)+2 x}\right ) \, dx+\frac {2}{5} \text {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (-\frac {7}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )-\frac {8}{5} \int \left (\frac {3 \sqrt [3]{-2+x^2+2 x^3}}{\left (2+2 x+x^2\right )^2}+\frac {x \sqrt [3]{-2+x^2+2 x^3}}{\left (2+2 x+x^2\right )^2}\right ) \, dx-\frac {17}{5} \text {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{-\frac {7}{6}+x} \, dx,x,\frac {1}{6}+x\right )+6 \text {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (-\frac {1}{6}+x\right )^5} \, dx,x,\frac {1}{6}+x\right )+9 \text {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (-\frac {1}{6}+x\right )^2} \, dx,x,\frac {1}{6}+x\right )-\text {Subst}\left (\int \frac {\sqrt [3]{-\frac {107}{54}-\frac {x}{6}+2 x^3}}{\left (-\frac {1}{6}+x\right )^3} \, dx,x,\frac {1}{6}+x\right ) \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\frac {\sqrt [3]{-2+x^2+2 x^3} \left (12-12 x^2+54 x^3+3 x^4-27 x^5-38 x^6\right )}{4 x^4 \left (-2+x^2+x^3\right )}-\frac {7 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-2+x^2+2 x^3}}\right )}{\sqrt {3}}-\frac {7}{3} \log \left (-x+\sqrt [3]{-2+x^2+2 x^3}\right )+\frac {7}{6} \log \left (x^2+x \sqrt [3]{-2+x^2+2 x^3}+\left (-2+x^2+2 x^3\right )^{2/3}\right ) \]
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Time = 3.16 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.28
method | result | size |
pseudoelliptic | \(\frac {\left (114 x^{6}+81 x^{5}-9 x^{4}-162 x^{3}+36 x^{2}-36\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}-14 x^{4} \left (x^{3}+x^{2}-2\right ) \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}\right )}{3 x}\right )+\ln \left (\frac {x^{2}+x \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}+\left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-x +\left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}}{x}\right )\right )}{12 \left (x^{2}+x \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}+\left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}}\right ) x^{4} \left (x -\left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}\right )}\) | \(199\) |
trager | \(-\frac {\left (38 x^{6}+27 x^{5}-3 x^{4}-54 x^{3}+12 x^{2}-12\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}}}{4 \left (x^{3}+x^{2}-2\right ) x^{4}}-\frac {7 \ln \left (-\frac {20044800 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{3}+468480 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x -1586400 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}-80179200 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{2}+1292640 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{3}-3305 \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x +2329 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} x^{2}+383520 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}+2490 x^{3}+160358400 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2}+1079 x^{2}-767040 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )-2158}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )}{3}+1120 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) \ln \left (-\frac {149196441600 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{3}+4089968640 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x +2476821600 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}-596785766400 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2} x^{2}-9892153440 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{3}+5160045 \left (2 x^{3}+x^{2}-2\right )^{\frac {2}{3}} x -13680813 \left (2 x^{3}+x^{2}-2\right )^{\frac {1}{3}} x^{2}-1771233600 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right ) x^{2}+16568849 x^{3}+1193571532800 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )^{2}+4810311 x^{2}+3542467200 \operatorname {RootOf}\left (230400 \textit {\_Z}^{2}-480 \textit {\_Z} +1\right )-9620622}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )\) | \(495\) |
risch | \(\text {Expression too large to display}\) | \(1291\) |
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Time = 1.01 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\frac {28 \, \sqrt {3} {\left (x^{7} + x^{6} - 2 \, x^{4}\right )} \arctan \left (\frac {1078 \, \sqrt {3} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} x^{2} + 196 \, \sqrt {3} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (669 \, x^{3} + 32 \, x^{2} - 64\right )}}{1315 \, x^{3} - 8 \, x^{2} + 16}\right ) - 14 \, {\left (x^{7} + x^{6} - 2 \, x^{4}\right )} \log \left (\frac {x^{3} + 3 \, {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} x^{2} + x^{2} - 3 \, {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {2}{3}} x - 2}{x^{3} + x^{2} - 2}\right ) - 3 \, {\left (38 \, x^{6} + 27 \, x^{5} - 3 \, x^{4} - 54 \, x^{3} + 12 \, x^{2} - 12\right )} {\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}}}{12 \, {\left (x^{7} + x^{6} - 2 \, x^{4}\right )}} \]
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\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int \frac {\left (x + 1\right ) \left (x^{2} - 6\right ) \left (x^{2} - 2\right ) \left (x^{2} - 2 x + 2\right ) \sqrt [3]{2 x^{3} + x^{2} - 2}}{x^{5} \left (x - 1\right )^{2} \left (x^{2} + 2 x + 2\right )^{2}}\, dx \]
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\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int { \frac {{\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} + 2\right )} {\left (x^{2} - 2\right )} {\left (x^{2} - 6\right )}}{{\left (x^{3} + x^{2} - 2\right )}^{2} x^{5}} \,d x } \]
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\[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int { \frac {{\left (2 \, x^{3} + x^{2} - 2\right )}^{\frac {1}{3}} {\left (x^{3} - x^{2} + 2\right )} {\left (x^{2} - 2\right )} {\left (x^{2} - 6\right )}}{{\left (x^{3} + x^{2} - 2\right )}^{2} x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\left (-6+x^2\right ) \left (-2+x^2\right ) \left (2-x^2+x^3\right ) \sqrt [3]{-2+x^2+2 x^3}}{x^5 \left (-2+x^2+x^3\right )^2} \, dx=\int \frac {\left (x^2-2\right )\,\left (x^2-6\right )\,\left (x^3-x^2+2\right )\,{\left (2\,x^3+x^2-2\right )}^{1/3}}{x^5\,{\left (x^3+x^2-2\right )}^2} \,d x \]
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