Optimal. Leaf size=20 \[ \frac {3 \left (x^4-x^2\right )^{4/3}}{4 x^4} \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, 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^4-x^2\right )^{4/3}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 1590
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt [3]{-x^2+x^4}}{x^3} \, dx &=\frac {3 \left (-x^2+x^4\right )^{4/3}}{4 x^4}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \begin {gather*} \frac {3 \left (x^2 \left (x^2-1\right )\right )^{4/3}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 20, normalized size = 1.00 \begin {gather*} \frac {3 \left (-x^2+x^4\right )^{4/3}}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 21, normalized size = 1.05 \begin {gather*} \frac {3 \, {\left (x^{4} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} - 1\right )}}{4 \, x^{2}} \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^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 22, normalized size = 1.10
method | result | size |
trager | \(\frac {3 \left (x^{2}-1\right ) \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{4 x^{2}}\) | \(22\) |
gosper | \(\frac {3 \left (-1+x \right ) \left (1+x \right ) \left (x^{4}-x^{2}\right )^{\frac {1}{3}}}{4 x^{2}}\) | \(23\) |
risch | \(\frac {3 \left (x^{2} \left (x^{2}-1\right )\right )^{\frac {1}{3}} \left (x^{4}-2 x^{2}+1\right )}{4 x^{2} \left (x^{2}-1\right )}\) | \(34\) |
meijerg | \(-\frac {3 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {2}{3}, -\frac {1}{3}\right ], \left [\frac {1}{3}\right ], x^{2}\right )}{4 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}} x^{\frac {4}{3}}}+\frac {3 \mathrm {signum}\left (x^{2}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{2}\right ) x^{\frac {2}{3}}}{2 \left (-\mathrm {signum}\left (x^{2}-1\right )\right )^{\frac {1}{3}}}\) | \(66\) |
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^{4} - x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{x^{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.21, size = 21, normalized size = 1.05 \begin {gather*} \frac {3\,\left (x^2-1\right )\,{\left (x^4-x^2\right )}^{1/3}}{4\,x^2} \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^{2} \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 1\right )}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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