Optimal. Leaf size=104 \[ \frac {\left (x^3-1\right )^{2/3}}{4 x^2}-\frac {1}{6} \text {RootSum}\left [4 \text {$\#$1}^6-7 \text {$\#$1}^3+2\& ,\frac {-\text {$\#$1}^3 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )+\text {$\#$1}^3 \log (x)+2 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )-2 \log (x)}{8 \text {$\#$1}^4-7 \text {$\#$1}}\& \right ] \]
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Rubi [B] time = 0.62, antiderivative size = 211, normalized size of antiderivative = 2.03, number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6728, 277, 239, 430, 429} \begin {gather*} \frac {\left (17+5 \sqrt {17}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1-\sqrt {17}}\right )}{34 \left (1-\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (17-5 \sqrt {17}\right ) x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1+\sqrt {17}}\right )}{34 \left (1+\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {1}{4} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {\left (x^3-1\right )^{2/3}}{4 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 239
Rule 277
Rule 429
Rule 430
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^3 \left (-4+x^3+x^6\right )} \, dx &=\int \left (-\frac {\left (-1+x^3\right )^{2/3}}{2 x^3}+\frac {\left (-1+x^3\right )^{2/3} \left (3+x^3\right )}{2 \left (-4+x^3+x^6\right )}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{x^3} \, dx\right )+\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3} \left (3+x^3\right )}{-4+x^3+x^6} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {1}{2} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx+\frac {1}{2} \int \left (\frac {\left (1+\frac {5}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+2 x^3}+\frac {\left (1-\frac {5}{\sqrt {17}}\right ) \left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+2 x^3}\right ) \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{34} \left (17-5 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1+\sqrt {17}+2 x^3} \, dx+\frac {1}{34} \left (17+5 \sqrt {17}\right ) \int \frac {\left (-1+x^3\right )^{2/3}}{1-\sqrt {17}+2 x^3} \, dx\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {\left (\left (17-5 \sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1+\sqrt {17}+2 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}+\frac {\left (\left (17+5 \sqrt {17}\right ) \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1-\sqrt {17}+2 x^3} \, dx}{34 \left (1-x^3\right )^{2/3}}\\ &=\frac {\left (-1+x^3\right )^{2/3}}{4 x^2}+\frac {\left (17+5 \sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1-\sqrt {17}}\right )}{34 \left (1-\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (17-5 \sqrt {17}\right ) x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {2 x^3}{1+\sqrt {17}}\right )}{34 \left (1+\sqrt {17}\right ) \left (1-x^3\right )^{2/3}}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{2 \sqrt {3}}+\frac {1}{4} \log \left (-x+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [B] time = 0.75, size = 387, normalized size = 3.72 \begin {gather*} \frac {\left (x^3-1\right )^{2/3}}{4 x^2}+\frac {-2 \sqrt [3]{199-47 \sqrt {17}} \log \left (\sqrt [3]{7-\sqrt {17}}-\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}\right )+2 \sqrt [3]{199+47 \sqrt {17}} \log \left (\sqrt [3]{7+\sqrt {17}}-\frac {2^{2/3} x}{\sqrt [3]{x^3-1}}\right )-2 \sqrt {3} \sqrt [3]{199+47 \sqrt {17}} \tan ^{-1}\left (\frac {\frac {2\ 2^{2/3} x}{\sqrt [3]{7+\sqrt {17}} \sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )+2 \sqrt {3} \sqrt [3]{199-47 \sqrt {17}} \tan ^{-1}\left (\frac {\frac {2\ 2^{2/3} x}{\sqrt [3]{-\left (\left (\sqrt {17}-7\right ) \left (x^3-1\right )\right )}}+1}{\sqrt {3}}\right )+\sqrt [3]{199-47 \sqrt {17}} \log \left (\frac {2^{2/3} \sqrt [3]{7-\sqrt {17}} x}{\sqrt [3]{x^3-1}}+\frac {2 \sqrt [3]{2} x^2}{\left (x^3-1\right )^{2/3}}+\left (7-\sqrt {17}\right )^{2/3}\right )-\sqrt [3]{199+47 \sqrt {17}} \log \left (\frac {2^{2/3} \sqrt [3]{7+\sqrt {17}} x}{\sqrt [3]{x^3-1}}+\frac {2 \sqrt [3]{2} x^2}{\left (x^3-1\right )^{2/3}}+\left (7+\sqrt {17}\right )^{2/3}\right )}{24\ 2^{2/3} \sqrt {17}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.20, size = 104, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{2/3}}{4 x^2}-\frac {1}{6} \text {RootSum}\left [2-7 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-2 \log (x)+2 \log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-7 \text {$\#$1}+8 \text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + x^{3} - 4\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 275.84, size = 7989, normalized size = 76.82
method | result | size |
risch | \(\text {Expression too large to display}\) | \(7989\) |
trager | \(\text {Expression too large to display}\) | \(10402\) |
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} + 2\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + x^{3} - 4\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^3-1\right )}^{2/3}\,\left (x^3+2\right )}{x^3\,\left (x^6+x^3-4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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