Optimal. Leaf size=18 \[ -\frac {3 x^2}{2 \left (x^4-x\right )^{2/3}} \]
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Rubi [A] time = 0.07, antiderivative size = 25, normalized size of antiderivative = 1.39, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2056, 449} \begin {gather*} \frac {3 x \sqrt [3]{x^4-x}}{2 \left (1-x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (2+x^3\right ) \sqrt [3]{-x+x^4}}{\left (-1+x^3\right )^2} \, dx &=\frac {\sqrt [3]{-x+x^4} \int \frac {\sqrt [3]{x} \left (2+x^3\right )}{\left (-1+x^3\right )^{5/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^3}}\\ &=\frac {3 x \sqrt [3]{-x+x^4}}{2 \left (1-x^3\right )}\\ \end {align*}
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Mathematica [C] time = 0.05, size = 61, normalized size = 3.39 \begin {gather*} \frac {3 \sqrt [3]{x \left (x^3-1\right )} \left (13 x \, _2F_1\left (\frac {4}{9},\frac {5}{3};\frac {13}{9};x^3\right )+2 x^4 \, _2F_1\left (\frac {13}{9},\frac {5}{3};\frac {22}{9};x^3\right )\right )}{26 \sqrt [3]{1-x^3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.20, size = 18, normalized size = 1.00 \begin {gather*} -\frac {3 x^2}{2 \left (x^4-x\right )^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 19, normalized size = 1.06 \begin {gather*} -\frac {3 \, {\left (x^{4} - x\right )}^{\frac {1}{3}} x}{2 \, {\left (x^{3} - 1\right )}} \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^{4} - x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{{\left (x^{3} - 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 20, normalized size = 1.11 \begin {gather*} -\frac {3 x \left (x^{4}-x \right )^{\frac {1}{3}}}{2 \left (x^{3}-1\right )} \end {gather*}
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^{4} - x\right )}^{\frac {1}{3}} {\left (x^{3} + 2\right )}}{{\left (x^{3} - 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 21, normalized size = 1.17 \begin {gather*} -\frac {3\,x\,{\left (x^4-x\right )}^{1/3}}{2\,\left (x^3-1\right )} \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 {\sqrt [3]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x^{3} + 2\right )}{\left (x - 1\right )^{2} \left (x^{2} + x + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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