3.2375 \(\int \frac{1}{\left (1+\sqrt [3]{x}\right ) x^{3/2}} \, dx\)

Optimal. Leaf size=23 \[ \frac{6}{\sqrt [6]{x}}-\frac{2}{\sqrt{x}}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0314003, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{6}{\sqrt [6]{x}}-\frac{2}{\sqrt{x}}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 4.85307, size = 20, normalized size = 0.87 \[ 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} - \frac{2}{\sqrt{x}} + \frac{6}{\sqrt [6]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0195794, size = 23, normalized size = 1. \[ \frac{6}{\sqrt [6]{x}}-\frac{2}{\sqrt{x}}+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 18, normalized size = 0.8 \[ 6\,{\frac{1}{\sqrt [6]{x}}}+6\,\arctan \left ( \sqrt [6]{x} \right ) -2\,{\frac{1}{\sqrt{x}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x^(1/3))/x^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.599, size = 26, normalized size = 1.13 \[ \frac{2 \,{\left (3 \, x^{\frac{1}{3}} - 1\right )}}{\sqrt{x}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^(3/2)*(x^(1/3) + 1)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.223659, size = 28, normalized size = 1.22 \[ \frac{2 \,{\left (3 \, \sqrt{x} \arctan \left (x^{\frac{1}{6}}\right ) + 3 \, x^{\frac{1}{3}} - 1\right )}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^(3/2)*(x^(1/3) + 1)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.54313, size = 20, normalized size = 0.87 \[ 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} - \frac{2}{\sqrt{x}} + \frac{6}{\sqrt [6]{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218293, size = 26, normalized size = 1.13 \[ \frac{2 \,{\left (3 \, x^{\frac{1}{3}} - 1\right )}}{\sqrt{x}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^(3/2)*(x^(1/3) + 1)),x, algorithm="giac")

[Out]

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