Optimal. Leaf size=18 \[ \frac {3 \left (x^4+x^2\right )^{2/3}}{2 x^2} \]
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Rubi [A] time = 0.07, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2034, 763} \begin {gather*} \frac {3 \left (x^4+x^2\right )^{2/3}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 763
Rule 2034
Rubi steps
\begin {align*} \int \frac {-1+x^2}{x \sqrt [3]{x^2+x^4}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {-1+x}{x \sqrt [3]{x+x^2}} \, dx,x,x^2\right )\\ &=\frac {3 \left (x^2+x^4\right )^{2/3}}{2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.11 \begin {gather*} \frac {3 \left (x^2+1\right )}{2 \sqrt [3]{x^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 )^{2/3}}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 14, normalized size = 0.78 \begin {gather*} \frac {3 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{2 \, 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 {x^{2} - 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 15, normalized size = 0.83
method | result | size |
trager | \(\frac {3 \left (x^{4}+x^{2}\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(15\) |
gosper | \(\frac {\frac {3 x^{2}}{2}+\frac {3}{2}}{\left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\) | \(17\) |
risch | \(\frac {\frac {3 x^{2}}{2}+\frac {3}{2}}{\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{3}}}\) | \(19\) |
meijerg | \(\frac {3 \hypergeom \left (\left [-\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x^{2}\right )}{2 x^{\frac {2}{3}}}+\frac {3 \hypergeom \left (\left [\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{2}\right ) x^{\frac {4}{3}}}{4}\) | \(34\) |
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 {x^{2} - 1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 14, normalized size = 0.78 \begin {gather*} \frac {3\,{\left (x^4+x^2\right )}^{2/3}}{2\,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 {\left (x - 1\right ) \left (x + 1\right )}{x \sqrt [3]{x^{2} \left (x^{2} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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