Optimal. Leaf size=39 \[ \frac {3 x^{4/3}}{4}+\frac {3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {3 x^{4/3}}{4}+\frac {3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{2/3}}{1+\sqrt [3]{x}} \, dx &=3 \operatorname {Subst}\left (\int \frac {x^4}{1+x} \, dx,x,\sqrt [3]{x}\right )\\ &=3 \operatorname {Subst}\left (\int \left (-1+x-x^2+x^3+\frac {1}{1+x}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-3 \sqrt [3]{x}+\frac {3 x^{2/3}}{2}-x+\frac {3 x^{4/3}}{4}+3 \log \left (1+\sqrt [3]{x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 39, normalized size = 1.00 \[ \frac {3 x^{4/3}}{4}+\frac {3 x^{2/3}}{2}-x-3 \sqrt [3]{x}+3 \log \left (\sqrt [3]{x}+1\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.65, size = 25, normalized size = 0.64 \[ \frac {3}{4} \, {\left (x - 4\right )} x^{\frac {1}{3}} - x + \frac {3}{2} \, x^{\frac {2}{3}} + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 27, normalized size = 0.69 \[ \frac {3}{4} \, x^{\frac {4}{3}} - x + \frac {3}{2} \, x^{\frac {2}{3}} - 3 \, x^{\frac {1}{3}} + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 28, normalized size = 0.72 \[ \frac {3 x^{\frac {4}{3}}}{4}-x +3 \ln \left (x^{\frac {1}{3}}+1\right )+\frac {3 x^{\frac {2}{3}}}{2}-3 x^{\frac {1}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 42, normalized size = 1.08 \[ \frac {3}{4} \, {\left (x^{\frac {1}{3}} + 1\right )}^{4} - 4 \, {\left (x^{\frac {1}{3}} + 1\right )}^{3} + 9 \, {\left (x^{\frac {1}{3}} + 1\right )}^{2} - 12 \, x^{\frac {1}{3}} + 3 \, \log \left (x^{\frac {1}{3}} + 1\right ) - 12 \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.02, size = 27, normalized size = 0.69 \[ 3\,\ln \left (x^{1/3}+1\right )-x-3\,x^{1/3}+\frac {3\,x^{2/3}}{2}+\frac {3\,x^{4/3}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 34, normalized size = 0.87 \[ \frac {3 x^{\frac {4}{3}}}{4} + \frac {3 x^{\frac {2}{3}}}{2} - 3 \sqrt [3]{x} - x + 3 \log {\left (\sqrt [3]{x} + 1 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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