Optimal. Leaf size=18 \[ \frac {3 \left (x^4+x^2\right )^{4/3}}{4 x^4} \]
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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 = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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 = 23, normalized size = 1.28 \begin {gather*} \frac {3 \left (x^2+1\right ) \sqrt [3]{x^4+x^2}}{4 x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.07, size = 18, 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 = 0.47, size = 19, normalized size = 1.06 \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 = 0.08, size = 20, normalized size = 1.11
method | result | size |
gosper | \(\frac {3 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}{4 x^{2}}\) | \(20\) |
trager | \(\frac {3 \left (x^{2}+1\right ) \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}{4 x^{2}}\) | \(20\) |
meijerg | \(\frac {3 \hypergeom \left (\left [-\frac {2}{3}, -\frac {1}{3}\right ], \left [\frac {1}{3}\right ], -x^{2}\right )}{4 x^{\frac {4}{3}}}+\frac {3 \hypergeom \left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{2}\right ) x^{\frac {2}{3}}}{2}\) | \(34\) |
risch | \(\frac {3 \left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}} \left (x^{4}+2 x^{2}+1\right )}{4 x^{2} \left (x^{2}+1\right )}\) | \(34\) |
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 = 19, normalized size = 1.06 \begin {gather*} \frac {3\,{\left (x^4+x^2\right )}^{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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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x^{2} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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