Optimal. Leaf size=67 \[ 6 \sqrt [3]{\sqrt{x}+1}+3 \log \left (1-\sqrt [3]{\sqrt{x}+1}\right )-\frac{\log (x)}{2}-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{\sqrt{x}+1}+1}{\sqrt{3}}\right ) \]
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Rubi [A] time = 0.068901, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ 6 \sqrt [3]{\sqrt{x}+1}+3 \log \left (1-\sqrt [3]{\sqrt{x}+1}\right )-\frac{\log (x)}{2}-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{\sqrt{x}+1}+1}{\sqrt{3}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + Sqrt[x])^(1/3)/x,x]
[Out]
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Rubi in Sympy [A] time = 2.38318, size = 63, normalized size = 0.94 \[ 6 \sqrt [3]{\sqrt{x} + 1} - \log{\left (\sqrt{x} \right )} + 3 \log{\left (- \sqrt [3]{\sqrt{x} + 1} + 1 \right )} - 2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{\sqrt{x} + 1}}{3} + \frac{1}{3}\right ) \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+x**(1/2))**(1/3)/x,x)
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Mathematica [C] time = 0.0267403, size = 51, normalized size = 0.76 \[ \frac{-3 \left (\frac{1}{\sqrt{x}}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{1}{\sqrt{x}}\right )+6 \sqrt{x}+6}{\left (\sqrt{x}+1\right )^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 + Sqrt[x])^(1/3)/x,x]
[Out]
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Maple [A] time = 0.006, size = 64, normalized size = 1. \[ 6\,\sqrt [3]{1+\sqrt{x}}-\ln \left ( \left ( 1+\sqrt{x} \right ) ^{{\frac{2}{3}}}+\sqrt [3]{1+\sqrt{x}}+1 \right ) -2\,\arctan \left ( 1/3\, \left ( 1+2\,\sqrt [3]{1+\sqrt{x}} \right ) \sqrt{3} \right ) \sqrt{3}+2\,\ln \left ( \sqrt [3]{1+\sqrt{x}}-1 \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+x^(1/2))^(1/3)/x,x)
[Out]
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Maxima [A] time = 0.78451, size = 85, normalized size = 1.27 \[ -2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (\sqrt{x} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) + 6 \,{\left (\sqrt{x} + 1\right )}^{\frac{1}{3}} - \log \left ({\left (\sqrt{x} + 1\right )}^{\frac{2}{3}} +{\left (\sqrt{x} + 1\right )}^{\frac{1}{3}} + 1\right ) + 2 \, \log \left ({\left (\sqrt{x} + 1\right )}^{\frac{1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(x) + 1)^(1/3)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.287226, size = 85, normalized size = 1.27 \[ -2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (\sqrt{x} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) + 6 \,{\left (\sqrt{x} + 1\right )}^{\frac{1}{3}} - \log \left ({\left (\sqrt{x} + 1\right )}^{\frac{2}{3}} +{\left (\sqrt{x} + 1\right )}^{\frac{1}{3}} + 1\right ) + 2 \, \log \left ({\left (\sqrt{x} + 1\right )}^{\frac{1}{3}} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(x) + 1)^(1/3)/x,x, algorithm="fricas")
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Sympy [A] time = 3.86289, size = 39, normalized size = 0.58 \[ - \frac{2 \sqrt [6]{x} \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{\frac{e^{i \pi }}{\sqrt{x}}} \right )}}{\Gamma \left (\frac{2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+x**(1/2))**(1/3)/x,x)
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GIAC/XCAS [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((sqrt(x) + 1)^(1/3)/x,x, algorithm="giac")
[Out]