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

Optimal. Leaf size=41 \[ \frac{6 x^{7/6}}{7}-\frac{6 x^{5/6}}{5}+2 \sqrt{x}-6 \sqrt [6]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

[Out]

-6*x^(1/6) + 2*Sqrt[x] - (6*x^(5/6))/5 + (6*x^(7/6))/7 + 6*ArcTan[x^(1/6)]

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Rubi [A]  time = 0.0346666, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{6 x^{7/6}}{7}-\frac{6 x^{5/6}}{5}+2 \sqrt{x}-6 \sqrt [6]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(1 + x^(1/3)),x]

[Out]

-6*x^(1/6) + 2*Sqrt[x] - (6*x^(5/6))/5 + (6*x^(7/6))/7 + 6*ArcTan[x^(1/6)]

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Rubi in Sympy [A]  time = 2.88017, size = 37, normalized size = 0.9 \[ \frac{6 x^{\frac{7}{6}}}{7} - \frac{6 x^{\frac{5}{6}}}{5} - 6 \sqrt [6]{x} + 2 \sqrt{x} + 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/2)/(1+x**(1/3)),x)

[Out]

6*x**(7/6)/7 - 6*x**(5/6)/5 - 6*x**(1/6) + 2*sqrt(x) + 6*atan(x**(1/6))

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Mathematica [A]  time = 0.0132777, size = 41, normalized size = 1. \[ \frac{6 x^{7/6}}{7}-\frac{6 x^{5/6}}{5}+2 \sqrt{x}-6 \sqrt [6]{x}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(1 + x^(1/3)),x]

[Out]

-6*x^(1/6) + 2*Sqrt[x] - (6*x^(5/6))/5 + (6*x^(7/6))/7 + 6*ArcTan[x^(1/6)]

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Maple [A]  time = 0.002, size = 28, normalized size = 0.7 \[ -6\,\sqrt [6]{x}-{\frac{6}{5}{x}^{{\frac{5}{6}}}}+{\frac{6}{7}{x}^{{\frac{7}{6}}}}+6\,\arctan \left ( \sqrt [6]{x} \right ) +2\,\sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(1+x^(1/3)),x)

[Out]

-6*x^(1/6)-6/5*x^(5/6)+6/7*x^(7/6)+6*arctan(x^(1/6))+2*x^(1/2)

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Maxima [A]  time = 0.755433, size = 36, normalized size = 0.88 \[ \frac{6}{7} \, x^{\frac{7}{6}} - \frac{6}{5} \, x^{\frac{5}{6}} + 2 \, \sqrt{x} - 6 \, x^{\frac{1}{6}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^(1/3) + 1),x, algorithm="maxima")

[Out]

6/7*x^(7/6) - 6/5*x^(5/6) + 2*sqrt(x) - 6*x^(1/6) + 6*arctan(x^(1/6))

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Fricas [A]  time = 0.266904, size = 34, normalized size = 0.83 \[ \frac{6}{7} \,{\left (x - 7\right )} x^{\frac{1}{6}} - \frac{6}{5} \, x^{\frac{5}{6}} + 2 \, \sqrt{x} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^(1/3) + 1),x, algorithm="fricas")

[Out]

6/7*(x - 7)*x^(1/6) - 6/5*x^(5/6) + 2*sqrt(x) + 6*arctan(x^(1/6))

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Sympy [A]  time = 9.66581, size = 37, normalized size = 0.9 \[ \frac{6 x^{\frac{7}{6}}}{7} - \frac{6 x^{\frac{5}{6}}}{5} - 6 \sqrt [6]{x} + 2 \sqrt{x} + 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(1+x**(1/3)),x)

[Out]

6*x**(7/6)/7 - 6*x**(5/6)/5 - 6*x**(1/6) + 2*sqrt(x) + 6*atan(x**(1/6))

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GIAC/XCAS [A]  time = 0.262878, size = 36, normalized size = 0.88 \[ \frac{6}{7} \, x^{\frac{7}{6}} - \frac{6}{5} \, x^{\frac{5}{6}} + 2 \, \sqrt{x} - 6 \, x^{\frac{1}{6}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/(x^(1/3) + 1),x, algorithm="giac")

[Out]

6/7*x^(7/6) - 6/5*x^(5/6) + 2*sqrt(x) - 6*x^(1/6) + 6*arctan(x^(1/6))