Optimal. Leaf size=18 \[ \frac {3 \left (x^3-x\right )^{4/3}}{8 x^4} \]
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Rubi [A] time = 0.02, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2014} \begin {gather*} \frac {3 \left (x^3-x\right )^{4/3}}{8 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 2014
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-x+x^3}}{x^4} \, dx &=\frac {3 \left (-x+x^3\right )^{4/3}}{8 x^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {3 \left (x \left (x^2-1\right )\right )^{4/3}}{8 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 18, normalized size = 1.00 \begin {gather*} \frac {3 \left (-x+x^3\right )^{4/3}}{8 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 19, normalized size = 1.06 \begin {gather*} \frac {3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}}{8 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.33, size = 11, normalized size = 0.61 \begin {gather*} \frac {3}{8} \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 20, normalized size = 1.11
method | result | size |
trager | \(\frac {3 \left (x^{2}-1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{8 x^{3}}\) | \(20\) |
gosper | \(\frac {3 \left (-1+x \right ) \left (1+x \right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{8 x^{3}}\) | \(21\) |
risch | \(\frac {3 \left (x \left (x^{2}-1\right )\right )^{\frac {1}{3}} \left (x^{4}-2 x^{2}+1\right )}{8 x^{3} \left (x^{2}-1\right )}\) | \(32\) |
meijerg | \(-\frac {3 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \left (-x^{2}+1\right )^{\frac {4}{3}}}{8 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{\frac {8}{3}}}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 22, normalized size = 1.22 \begin {gather*} \frac {3 \, {\left (x^{3} - x\right )} {\left (x + 1\right )}^{\frac {1}{3}} {\left (x - 1\right )}^{\frac {1}{3}}}{8 \, x^{\frac {11}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 31, normalized size = 1.72 \begin {gather*} -\frac {3\,{\left (x^3-x\right )}^{1/3}-3\,x^2\,{\left (x^3-x\right )}^{1/3}}{8\,x^3} \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 )}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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