Optimal. Leaf size=79 \[ \frac {\left (x^3-1\right )^{5/3}}{5 x^5}-\frac {2}{3} \text {RootSum}\left [2 \text {$\#$1}^6-5 \text {$\#$1}^3+5\& ,\frac {\text {$\#$1}^2 \log \left (\sqrt [3]{x^3-1}-\text {$\#$1} x\right )-\text {$\#$1}^2 \log (x)}{4 \text {$\#$1}^3-5}\& \right ] \]
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Rubi [C] time = 0.49, antiderivative size = 161, normalized size of antiderivative = 2.04, number of steps used = 8, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6728, 264, 1428, 430, 429} \begin {gather*} -\frac {8 x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1-i \sqrt {15}}\right )}{\sqrt {15} \left (\sqrt {15}+i\right ) \left (1-x^3\right )^{2/3}}+\frac {8 x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1+i \sqrt {15}}\right )}{\sqrt {15} \left (-\sqrt {15}+i\right ) \left (1-x^3\right )^{2/3}}+\frac {\left (x^3-1\right )^{5/3}}{5 x^5} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 264
Rule 429
Rule 430
Rule 1428
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (2+x^3+2 x^6\right )} \, dx &=\int \left (\frac {\left (-1+x^3\right )^{2/3}}{x^6}-\frac {2 \left (-1+x^3\right )^{2/3}}{2+x^3+2 x^6}\right ) \, dx\\ &=-\left (2 \int \frac {\left (-1+x^3\right )^{2/3}}{2+x^3+2 x^6} \, dx\right )+\int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {(8 i) \int \frac {\left (-1+x^3\right )^{2/3}}{1-i \sqrt {15}+4 x^3} \, dx}{\sqrt {15}}-\frac {(8 i) \int \frac {\left (-1+x^3\right )^{2/3}}{1+i \sqrt {15}+4 x^3} \, dx}{\sqrt {15}}\\ &=\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (8 i \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1-i \sqrt {15}+4 x^3} \, dx}{\sqrt {15} \left (1-x^3\right )^{2/3}}-\frac {\left (8 i \left (-1+x^3\right )^{2/3}\right ) \int \frac {\left (1-x^3\right )^{2/3}}{1+i \sqrt {15}+4 x^3} \, dx}{\sqrt {15} \left (1-x^3\right )^{2/3}}\\ &=\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {8 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1-i \sqrt {15}}\right )}{\sqrt {15} \left (i+\sqrt {15}\right ) \left (1-x^3\right )^{2/3}}+\frac {8 x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {4 x^3}{1+i \sqrt {15}}\right )}{\sqrt {15} \left (i-\sqrt {15}\right ) \left (1-x^3\right )^{2/3}}\\ \end {align*}
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Mathematica [F] time = 0.15, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^3\right )^{2/3} \left (2+x^3\right )}{x^6 \left (2+x^3+2 x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.20, size = 79, normalized size = 1.00 \begin {gather*} \frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {2}{3} \text {RootSum}\left [5-5 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x) \text {$\#$1}^2+\log \left (\sqrt [3]{-1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^2}{-5+4 \text {$\#$1}^3}\&\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 (2 \, x^{6} + x^{3} + 2\right )} x^{6}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 280.11, size = 6506, normalized size = 82.35
method | result | size |
risch | \(\text {Expression too large to display}\) | \(6506\) |
trager | \(\text {Expression too large to display}\) | \(12148\) |
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 (2 \, x^{6} + x^{3} + 2\right )} x^{6}}\,{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^6\,\left (2\,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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