Optimal. Leaf size=25 \[ \frac {1}{40} \left (x^3+1\right )^{2/3} \left (5 x^6+2 x^3-3\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.08, 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 {1}{8} \left (x^3+1\right )^{8/3}-\frac {1}{5} \left (x^3+1\right )^{5/3} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^5 \left (1+x^3\right )^{2/3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int x (1+x)^{2/3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-(1+x)^{2/3}+(1+x)^{5/3}\right ) \, dx,x,x^3\right )\\ &=-\frac {1}{5} \left (1+x^3\right )^{5/3}+\frac {1}{8} \left (1+x^3\right )^{8/3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{40} \left (x^3+1\right )^{5/3} \left (5 x^3-3\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.02, size = 20, normalized size = 0.80 \begin {gather*} \frac {1}{40} \left (1+x^3\right )^{5/3} \left (-3+5 x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 21, normalized size = 0.84 \begin {gather*} \frac {1}{40} \, {\left (5 \, x^{6} + 2 \, x^{3} - 3\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{8} \, {\left (x^{3} + 1\right )}^{\frac {8}{3}} - \frac {1}{5} \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.06, size = 17, normalized size = 0.68
method | result | size |
meijerg | \(\frac {\hypergeom \left (\left [-\frac {2}{3}, 2\right ], \relax [3], -x^{3}\right ) x^{6}}{6}\) | \(17\) |
trager | \(\left (\frac {1}{8} x^{6}+\frac {1}{20} x^{3}-\frac {3}{40}\right ) \left (x^{3}+1\right )^{\frac {2}{3}}\) | \(21\) |
risch | \(\frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (5 x^{6}+2 x^{3}-3\right )}{40}\) | \(22\) |
gosper | \(\frac {\left (1+x \right ) \left (x^{2}-x +1\right ) \left (5 x^{3}-3\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{40}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 19, normalized size = 0.76 \begin {gather*} \frac {1}{8} \, {\left (x^{3} + 1\right )}^{\frac {8}{3}} - \frac {1}{5} \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 20, normalized size = 0.80 \begin {gather*} {\left (x^3+1\right )}^{2/3}\,\left (\frac {x^6}{8}+\frac {x^3}{20}-\frac {3}{40}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 37, normalized size = 1.48 \begin {gather*} \frac {x^{6} \left (x^{3} + 1\right )^{\frac {2}{3}}}{8} + \frac {x^{3} \left (x^{3} + 1\right )^{\frac {2}{3}}}{20} - \frac {3 \left (x^{3} + 1\right )^{\frac {2}{3}}}{40} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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