Optimal. Leaf size=73 \[ \frac {\left (x^3+1\right )^{2/3}}{2 x^2}-\frac {2}{3} \text {RootSum}\left [2 \text {$\#$1}^6-3 \text {$\#$1}^3+2\& ,\frac {\log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-\log (x)}{4 \text {$\#$1}^4-3 \text {$\#$1}}\& \right ] \]
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Rubi [C] time = 0.47, antiderivative size = 187, normalized size of antiderivative = 2.56, number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6728, 277, 239, 429} \begin {gather*} \frac {\left (3 \sqrt {7}+7 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {2 x^3}{1-i \sqrt {7}}\right )}{7 \left (\sqrt {7}+i\right )}+\frac {\left (-3 \sqrt {7}+7 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {2 x^3}{1+i \sqrt {7}}\right )}{7 \left (-\sqrt {7}+i\right )}+\frac {1}{2} \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 )}{\sqrt {3}}+\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 239
Rule 277
Rule 429
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (2+x^3+x^6\right )} \, dx &=\int \left (-\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{2+x^3+x^6}\right ) \, dx\\ &=-\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{2+x^3+x^6} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\int \left (\frac {\left (1-\frac {3 i}{\sqrt {7}}\right ) \left (1+x^3\right )^{2/3}}{1-i \sqrt {7}+2 x^3}+\frac {\left (1+\frac {3 i}{\sqrt {7}}\right ) \left (1+x^3\right )^{2/3}}{1+i \sqrt {7}+2 x^3}\right ) \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{7} \left (7-3 i \sqrt {7}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1-i \sqrt {7}+2 x^3} \, dx+\frac {1}{7} \left (7+3 i \sqrt {7}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1+i \sqrt {7}+2 x^3} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (7 i+3 \sqrt {7}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {2 x^3}{1-i \sqrt {7}}\right )}{7 \left (i+\sqrt {7}\right )}+\frac {\left (7 i-3 \sqrt {7}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {2 x^3}{1+i \sqrt {7}}\right )}{7 \left (i-\sqrt {7}\right )}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.21, size = 381, normalized size = 5.22 \begin {gather*} \frac {\left (x^3+1\right )^{2/3}}{2 x^2}+\frac {i \left (-\frac {2 \log \left (\sqrt [3]{\sqrt {7}+i}-\frac {\sqrt [3]{\sqrt {7}-i} x}{\sqrt [3]{x^3+1}}\right )}{\sqrt [3]{\frac {\sqrt {7}-i}{\sqrt {7}+i}}}+2 \sqrt [3]{\frac {\sqrt {7}-i}{\sqrt {7}+i}} \log \left (\sqrt [3]{\sqrt {7}-i}-\frac {\sqrt [3]{\sqrt {7}+i} x}{\sqrt [3]{x^3+1}}\right )+\frac {2 \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {\left (\sqrt {7}-i\right )^{2/3} x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )+\log \left (\frac {2 x}{\sqrt [3]{x^3+1}}+\frac {\left (\sqrt {7}-i\right )^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (\sqrt {7}+i\right )^{2/3}\right )\right )}{\left (\sqrt {7}-i\right )^{2/3}}-\frac {2 \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {\left (\sqrt {7}+i\right )^{2/3} x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )+\log \left (\frac {2 x}{\sqrt [3]{x^3+1}}+\frac {\left (\sqrt {7}+i\right )^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (\sqrt {7}-i\right )^{2/3}\right )\right )}{\left (\sqrt {7}+i\right )^{2/3}}\right )}{3 \sqrt {7}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.00, size = 73, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {2}{3} \text {RootSum}\left [2-3 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )}{-3 \text {$\#$1}+4 \text {$\#$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} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{6} + x^{3} + 2\right )} x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 161.74, size = 5531, normalized size = 75.77
method | result | size |
risch | \(\text {Expression too large to display}\) | \(5531\) |
trager | \(\text {Expression too large to display}\) | \(9663\) |
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} + 1\right )}^{\frac {2}{3}} {\left (x^{3} - 2\right )}}{{\left (x^{6} + x^{3} + 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^3+1\right )}^{2/3}\,\left (x^3-2\right )}{x^3\,\left (x^6+x^3+2\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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