Optimal. Leaf size=23 \[ \frac {3 \left (x^2-1\right ) \sqrt [3]{x^3-x}}{2 x} \]
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Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 0.78, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1590} \begin {gather*} \frac {3 \left (x^3-x\right )^{4/3}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin {align*} \int \frac {\left (1+3 x^2\right ) \sqrt [3]{-x+x^3}}{x^2} \, dx &=\frac {3 \left (-x+x^3\right )^{4/3}}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.78 \begin {gather*} \frac {3 \left (x \left (x^2-1\right )\right )^{4/3}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 18, normalized size = 0.78 \begin {gather*} \frac {3 \left (-x+x^3\right )^{4/3}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 19, normalized size = 0.83 \begin {gather*} \frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 26, normalized size = 1.13 \begin {gather*} \frac {3}{2} \, x^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - \frac {3}{2} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 20, normalized size = 0.87
method | result | size |
trager | \(\frac {3 \left (x^{2}-1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{2 x}\) | \(20\) |
gosper | \(\frac {3 \left (-1+x \right ) \left (1+x \right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{2 x}\) | \(21\) |
risch | \(\frac {3 \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}} \left (x^{4}-2 x^{2}+1\right )}{2 x \left (x^{2}-1\right )}\) | \(32\) |
meijerg | \(-\frac {3 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{2}\right )}{2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{\frac {2}{3}}}+\frac {9 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{2}\right ) x^{\frac {4}{3}}}{4 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}}}\) | \(66\) |
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} - x\right )}^{\frac {1}{3}} {\left (3 \, x^{2} + 1\right )}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.18, size = 19, normalized size = 0.83 \begin {gather*} \frac {3\,{\left (x^3-x\right )}^{1/3}\,\left (x^2-1\right )}{2\,x} \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 + 1\right )} \left (3 x^{2} + 1\right )}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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