Optimal. Leaf size=18 \[ \frac {3}{8} \left (x^2-3\right ) \sqrt [3]{x^2+1} \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.50, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {3}{8} \left (x^2+1\right )^{4/3}-\frac {3}{2} \sqrt [3]{x^2+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^3}{\left (1+x^2\right )^{2/3}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x}{(1+x)^{2/3}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {1}{(1+x)^{2/3}}+\sqrt [3]{1+x}\right ) \, dx,x,x^2\right )\\ &=-\frac {3}{2} \sqrt [3]{1+x^2}+\frac {3}{8} \left (1+x^2\right )^{4/3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {3}{8} \left (x^2-3\right ) \sqrt [3]{x^2+1} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {3}{8} \left (-3+x^2\right ) \sqrt [3]{1+x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 14, normalized size = 0.78 \begin {gather*} \frac {3}{8} \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 19, normalized size = 1.06 \begin {gather*} \frac {3}{8} \, {\left (x^{2} + 1\right )}^{\frac {4}{3}} - \frac {3}{2} \, {\left (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 = 0.05, size = 15, normalized size = 0.83
method | result | size |
gosper | \(\frac {3 \left (x^{2}-3\right ) \left (x^{2}+1\right )^{\frac {1}{3}}}{8}\) | \(15\) |
risch | \(\frac {3 \left (x^{2}-3\right ) \left (x^{2}+1\right )^{\frac {1}{3}}}{8}\) | \(15\) |
trager | \(\left (\frac {3 x^{2}}{8}-\frac {9}{8}\right ) \left (x^{2}+1\right )^{\frac {1}{3}}\) | \(16\) |
meijerg | \(\frac {\hypergeom \left (\left [\frac {2}{3}, 2\right ], \relax [3], -x^{2}\right ) x^{4}}{4}\) | \(17\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 19, normalized size = 1.06 \begin {gather*} \frac {3}{8} \, {\left (x^{2} + 1\right )}^{\frac {4}{3}} - \frac {3}{2} \, {\left (x^{2} + 1\right )}^{\frac {1}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.20, size = 14, normalized size = 0.78 \begin {gather*} \frac {3\,{\left (x^2+1\right )}^{1/3}\,\left (x^2-3\right )}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.78, size = 26, normalized size = 1.44 \begin {gather*} \frac {3 x^{2} \sqrt [3]{x^{2} + 1}}{8} - \frac {9 \sqrt [3]{x^{2} + 1}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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