3.809 \(\int \frac{\sqrt [3]{1+\sqrt{x}}}{x} \, dx\)

Optimal. Leaf size=67 \[ 6 \sqrt [3]{\sqrt{x}+1}+3 \log \left (1-\sqrt [3]{\sqrt{x}+1}\right )-\frac{\log (x)}{2}-2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{\sqrt{x}+1}+1}{\sqrt{3}}\right ) \]

[Out]

6*(1 + Sqrt[x])^(1/3) - 2*Sqrt[3]*ArcTan[(1 + 2*(1 + Sqrt[x])^(1/3))/Sqrt[3]] +
3*Log[1 - (1 + Sqrt[x])^(1/3)] - Log[x]/2

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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]

6*(1 + Sqrt[x])^(1/3) - 2*Sqrt[3]*ArcTan[(1 + 2*(1 + Sqrt[x])^(1/3))/Sqrt[3]] +
3*Log[1 - (1 + Sqrt[x])^(1/3)] - Log[x]/2

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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)

[Out]

6*(sqrt(x) + 1)**(1/3) - log(sqrt(x)) + 3*log(-(sqrt(x) + 1)**(1/3) + 1) - 2*sqr
t(3)*atan(sqrt(3)*(2*(sqrt(x) + 1)**(1/3)/3 + 1/3))

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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]

(6 + 6*Sqrt[x] - 3*(1 + 1/Sqrt[x])^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, -(1/Sq
rt[x])])/(1 + Sqrt[x])^(2/3)

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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]

6*(1+x^(1/2))^(1/3)-ln((1+x^(1/2))^(2/3)+(1+x^(1/2))^(1/3)+1)-2*arctan(1/3*(1+2*
(1+x^(1/2))^(1/3))*3^(1/2))*3^(1/2)+2*ln((1+x^(1/2))^(1/3)-1)

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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]

-2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(sqrt(x) + 1)^(1/3) + 1)) + 6*(sqrt(x) + 1)^(1/
3) - log((sqrt(x) + 1)^(2/3) + (sqrt(x) + 1)^(1/3) + 1) + 2*log((sqrt(x) + 1)^(1
/3) - 1)

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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")

[Out]

-2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(sqrt(x) + 1)^(1/3) + 1)) + 6*(sqrt(x) + 1)^(1/
3) - log((sqrt(x) + 1)^(2/3) + (sqrt(x) + 1)^(1/3) + 1) + 2*log((sqrt(x) + 1)^(1
/3) - 1)

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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)

[Out]

-2*x**(1/6)*gamma(-1/3)*hyper((-1/3, -1/3), (2/3,), exp_polar(I*pi)/sqrt(x))/gam
ma(2/3)

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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]

Timed out