Optimal. Leaf size=151 \[ -\frac {3 \left (x^3-x^2+2\right )^{2/3}}{2 x^2}-2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3-x^2+2}-2 x\right )+\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^3-x^2+2} x+\sqrt [3]{2} \left (x^3-x^2+2\right )^{2/3}\right )}{\sqrt [3]{2}}+2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3-x^2+2}+x}\right ) \]
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Rubi [F] time = 2.88, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx &=\int \left (\frac {\left (2-x^2+x^3\right )^{2/3}}{1-x}+\frac {3 \left (2-x^2+x^3\right )^{2/3}}{x^3}+\frac {\left (2-x^2+x^3\right )^{2/3}}{x}+\frac {\left (2-x^2+x^3\right )^{2/3}}{2+2 x+x^2}\right ) \, dx\\ &=3 \int \frac {\left (2-x^2+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {\left (2-x^2+x^3\right )^{2/3}}{1-x} \, dx+\int \frac {\left (2-x^2+x^3\right )^{2/3}}{x} \, dx+\int \frac {\left (2-x^2+x^3\right )^{2/3}}{2+2 x+x^2} \, dx\\ &=3 \operatorname {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{2/3}}{\left (\frac {1}{3}+x\right )^3} \, dx,x,-\frac {1}{3}+x\right )+\int \left (\frac {i \left (2-x^2+x^3\right )^{2/3}}{(-2+2 i)-2 x}+\frac {i \left (2-x^2+x^3\right )^{2/3}}{(2+2 i)+2 x}\right ) \, dx+\operatorname {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{2/3}}{\frac {2}{3}-x} \, dx,x,-\frac {1}{3}+x\right )+\operatorname {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{2/3}}{\frac {1}{3}+x} \, dx,x,-\frac {1}{3}+x\right )\\ &=i \int \frac {\left (2-x^2+x^3\right )^{2/3}}{(-2+2 i)-2 x} \, dx+i \int \frac {\left (2-x^2+x^3\right )^{2/3}}{(2+2 i)+2 x} \, dx+\frac {\left (3 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\frac {2}{3}-x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (3 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\frac {1}{3}+x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (9 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\left (\frac {1}{3}+x\right )^3} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}\\ &=i \operatorname {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{2/3}}{\left (-\frac {8}{3}+2 i\right )-2 x} \, dx,x,-\frac {1}{3}+x\right )+i \operatorname {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{2/3}}{\left (\frac {8}{3}+2 i\right )+2 x} \, dx,x,-\frac {1}{3}+x\right )+\frac {\left (3 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\frac {2}{3}-x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (3 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\frac {1}{3}+x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (9 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\left (\frac {1}{3}+x\right )^3} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}\\ &=\frac {\left (3 i \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\left (-\frac {8}{3}+2 i\right )-2 x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (3 i \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\left (\frac {8}{3}+2 i\right )+2 x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (3 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\frac {2}{3}-x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (3 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\frac {1}{3}+x} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}+\frac {\left (9 \sqrt [3]{3} \left (2-x^2+x^3\right )^{2/3}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{2/3} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{2/3}}{\left (\frac {1}{3}+x\right )^3} \, dx,x,-\frac {1}{3}+x\right )}{\left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x\right )^{2/3} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}+9 \left (-\frac {1}{3}+x\right )^2-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) (-1+3 x)}{\sqrt [3]{26-15 \sqrt {3}}}\right )^{2/3}}\\ \end {align*}
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Mathematica [F] time = 0.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-6+x^2\right ) \left (2-x^2+x^3\right )^{2/3}}{x^3 \left (-2+x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.48, size = 151, normalized size = 1.00 \begin {gather*} -\frac {3 \left (2-x^2+x^3\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{2-x^2+x^3}}\right )-2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{2-x^2+x^3}\right )+\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{2-x^2+x^3}+\sqrt [3]{2} \left (2-x^2+x^3\right )^{2/3}\right )}{\sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 16.01, size = 420, normalized size = 2.78 \begin {gather*} -\frac {2 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {3 \, \sqrt {3} \left (-4\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{6} - x^{5} - 8 \, x^{4} + 4 \, x^{3} - 4 \, x\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} + 6 \, \sqrt {3} \left (-4\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{7} + x^{6} + 32 \, x^{5} - 4 \, x^{4} + 4 \, x^{2}\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} - 111 \, x^{8} + 33 \, x^{7} + 221 \, x^{6} - 132 \, x^{5} + 6 \, x^{4} + 132 \, x^{3} - 12 \, x^{2} + 8\right )}}{3 \, {\left (109 \, x^{9} - 105 \, x^{8} + 3 \, x^{7} + 211 \, x^{6} - 12 \, x^{5} - 6 \, x^{4} + 12 \, x^{3} + 12 \, x^{2} - 8\right )}}\right ) - 2 \, \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {3 \, \left (-4\right )^{\frac {2}{3}} {\left (x^{3} - x^{2} + 2\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} x + \left (-4\right )^{\frac {1}{3}} {\left (x^{3} + x^{2} - 2\right )}}{x^{3} + x^{2} - 2}\right ) + \left (-4\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \, \left (-4\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x^{3} + 2 \, x\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} - \left (-4\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{5} + x^{4} + 32 \, x^{3} - 4 \, x^{2} + 4\right )} - 24 \, {\left (2 \, x^{5} - x^{4} + 2 \, x^{2}\right )} {\left (x^{3} - x^{2} + 2\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{5} + x^{4} - 4 \, x^{3} - 4 \, x^{2} + 4}\right ) + 9 \, {\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}}}{6 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} {\left (x^{2} - 6\right )}}{{\left (x^{3} + x^{2} - 2\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 19.80, size = 735, normalized size = 4.87
method | result | size |
risch | \(-\frac {3 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}}}{2 x^{2}}+\RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (\frac {\RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+2 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +2 \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}-\left (x^{3}-x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-9 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x^{3}-5 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}} x +\RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}+3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x^{2}-2 \RootOf \left (\textit {\_Z}^{3}+4\right )-6 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )+2 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \ln \left (-\frac {3 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}+4 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +4 \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+10 \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x^{3}+2 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}} x -3 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}-4 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right ) x^{2}+6 \RootOf \left (\textit {\_Z}^{3}+4\right )+8 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+2 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+4 \textit {\_Z}^{2}\right )}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )\) | \(735\) |
trager | \(-\frac {3 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}}}{2 x^{2}}+\RootOf \left (\textit {\_Z}^{3}+4\right ) \ln \left (\frac {827472528 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}-66909540 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}-3309890112 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}-1753973568 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +267638160 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{2}-1528686900 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}-146164464 \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+6619780224 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2}-535276320 \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right )+1723901100 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) x^{3}-139394875 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}-206868132 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) x^{2}+164938353 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}} x +16727385 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}+413736264 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right )-33454770 \RootOf \left (\textit {\_Z}^{3}+4\right )}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )+24 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) \ln \left (\frac {3211657920 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{3}-68956044 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{3}-12846631680 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+3507947136 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x +275824176 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} x^{2}+3958520472 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}+292328928 \left (x^{3}-x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2} x^{2}+25693263360 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}+4\right )^{2}-551648352 \RootOf \left (\textit {\_Z}^{3}+4\right )^{3} \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right )-7226230320 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) x^{3}+155151099 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{3}+2944019760 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right ) x^{2}-254781150 \left (x^{3}-x^{2}+2\right )^{\frac {2}{3}} x -63209707 \RootOf \left (\textit {\_Z}^{3}+4\right ) x^{2}-5888039520 \RootOf \left (\RootOf \left (\textit {\_Z}^{3}+4\right )^{2}+24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}+4\right )+576 \textit {\_Z}^{2}\right )+126419414 \RootOf \left (\textit {\_Z}^{3}+4\right )}{\left (-1+x \right ) \left (x^{2}+2 x +2\right )}\right )\) | \(1028\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} - x^{2} + 2\right )}^{\frac {2}{3}} {\left (x^{2} - 6\right )}}{{\left (x^{3} + x^{2} - 2\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^2-6\right )\,{\left (x^3-x^2+2\right )}^{2/3}}{x^3\,\left (x^3+x^2-2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x + 1\right ) \left (x^{2} - 2 x + 2\right )\right )^{\frac {2}{3}} \left (x^{2} - 6\right )}{x^{3} \left (x - 1\right ) \left (x^{2} + 2 x + 2\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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