Optimal. Leaf size=16 \[ -\frac {3 x^2}{2 \left (x+x^4\right )^{2/3}} \]
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Rubi [A]
time = 0.04, antiderivative size = 21, normalized size of antiderivative = 1.31, number of steps
used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2081, 460}
\begin {gather*} -\frac {3 x \sqrt [3]{x^4+x}}{2 \left (x^3+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 460
Rule 2081
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 [A]
time = 5.92, size = 21, normalized size = 1.31 \begin {gather*} -\frac {3 x \sqrt [3]{x+x^4}}{2 \left (1+x^3\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 18, normalized size = 1.12
method | result | size |
gosper | \(-\frac {3 x \left (x^{4}+x \right )^{\frac {1}{3}}}{2 \left (x^{3}+1\right )}\) | \(18\) |
trager | \(-\frac {3 x \left (x^{4}+x \right )^{\frac {1}{3}}}{2 \left (x^{3}+1\right )}\) | \(18\) |
risch | \(-\frac {3 \left (x \left (x^{3}+1\right )\right )^{\frac {1}{3}} x}{2 \left (x^{3}+1\right )}\) | \(20\) |
meijerg | \(-\frac {3 x^{\frac {4}{3}} \hypergeom \left (\left [\frac {4}{9}, \frac {5}{3}\right ], \left [\frac {13}{9}\right ], -x^{3}\right )}{2}+\frac {3 x^{\frac {13}{3}} \hypergeom \left (\left [\frac {13}{9}, \frac {5}{3}\right ], \left [\frac {22}{9}\right ], -x^{3}\right )}{13}\) | \(34\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 17, 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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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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 19, normalized size = 1.19 \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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