3.417 \(\int \frac{\sqrt [3]{x}}{\sqrt [4]{x}+\sqrt{x}} \, dx\)

Optimal. Leaf size=76 \[ \frac{6 x^{5/6}}{5}-\frac{12 x^{7/12}}{7}+3 \sqrt [3]{x}-12 \sqrt [12]{x}+6 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [4]{x}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [12]{x}}{\sqrt{3}}\right ) \]

[Out]

-12*x^(1/12) + 3*x^(1/3) - (12*x^(7/12))/7 + (6*x^(5/6))/5 - 4*Sqrt[3]*ArcTan[(1
- 2*x^(1/12))/Sqrt[3]] + 6*Log[1 + x^(1/12)] - 2*Log[1 + x^(1/4)]

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Rubi [A]  time = 0.0758517, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368 \[ \frac{6 x^{5/6}}{5}-\frac{12 x^{7/12}}{7}+3 \sqrt [3]{x}-12 \sqrt [12]{x}+6 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [4]{x}+1\right )-4 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [12]{x}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^(1/3)/(x^(1/4) + Sqrt[x]),x]

[Out]

-12*x^(1/12) + 3*x^(1/3) - (12*x^(7/12))/7 + (6*x^(5/6))/5 - 4*Sqrt[3]*ArcTan[(1
- 2*x^(1/12))/Sqrt[3]] + 6*Log[1 + x^(1/12)] - 2*Log[1 + x^(1/4)]

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Rubi in Sympy [A]  time = 3.87906, size = 75, normalized size = 0.99 \[ - \frac{12 x^{\frac{7}{12}}}{7} - 12 \sqrt [12]{x} + \frac{6 x^{\frac{5}{6}}}{5} + 3 \sqrt [3]{x} + 6 \log{\left (\sqrt [12]{x} + 1 \right )} - 2 \log{\left (\sqrt [4]{x} + 1 \right )} + 4 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [12]{x}}{3} - \frac{1}{3}\right ) \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(1/3)/(x**(1/4)+x**(1/2)),x)

[Out]

-12*x**(7/12)/7 - 12*x**(1/12) + 6*x**(5/6)/5 + 3*x**(1/3) + 6*log(x**(1/12) + 1
) - 2*log(x**(1/4) + 1) + 4*sqrt(3)*atan(sqrt(3)*(2*x**(1/12)/3 - 1/3))

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Mathematica [A]  time = 0.0260146, size = 83, normalized size = 1.09 \[ \frac{6 x^{5/6}}{5}-\frac{12 x^{7/12}}{7}+3 \sqrt [3]{x}-12 \sqrt [12]{x}+4 \log \left (\sqrt [12]{x}+1\right )-2 \log \left (\sqrt [6]{x}-\sqrt [12]{x}+1\right )+4 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [12]{x}-1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^(1/3)/(x^(1/4) + Sqrt[x]),x]

[Out]

-12*x^(1/12) + 3*x^(1/3) - (12*x^(7/12))/7 + (6*x^(5/6))/5 + 4*Sqrt[3]*ArcTan[(-
1 + 2*x^(1/12))/Sqrt[3]] + 4*Log[1 + x^(1/12)] - 2*Log[1 - x^(1/12) + x^(1/6)]

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Maple [A]  time = 0.006, size = 61, normalized size = 0.8 \[{\frac{6}{5}{x}^{{\frac{5}{6}}}}-{\frac{12}{7}{x}^{{\frac{7}{12}}}}+3\,\sqrt [3]{x}-12\,{x}^{1/12}+4\,\ln \left ( 1+{x}^{1/12} \right ) -2\,\ln \left ( 1-{x}^{1/12}+\sqrt [6]{x} \right ) +4\,\sqrt{3}\arctan \left ( 1/3\, \left ( 2\,{x}^{1/12}-1 \right ) \sqrt{3} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/3)/(x^(1/4)+x^(1/2)),x)

[Out]

6/5*x^(5/6)-12/7*x^(7/12)+3*x^(1/3)-12*x^(1/12)+4*ln(1+x^(1/12))-2*ln(1-x^(1/12)
+x^(1/6))+4*3^(1/2)*arctan(1/3*(2*x^(1/12)-1)*3^(1/2))

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Maxima [A]  time = 0.81326, size = 81, normalized size = 1.07 \[ 4 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{12}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{12}{7} \, x^{\frac{7}{12}} + 3 \, x^{\frac{1}{3}} - 12 \, x^{\frac{1}{12}} - 2 \, \log \left (x^{\frac{1}{6}} - x^{\frac{1}{12}} + 1\right ) + 4 \, \log \left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(1/3)/(sqrt(x) + x^(1/4)),x, algorithm="maxima")

[Out]

4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 6/5*x^(5/6) - 12/7*x^(7/12) + 3
*x^(1/3) - 12*x^(1/12) - 2*log(x^(1/6) - x^(1/12) + 1) + 4*log(x^(1/12) + 1)

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Fricas [A]  time = 0.281759, size = 81, normalized size = 1.07 \[ 4 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{12}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{12}{7} \, x^{\frac{7}{12}} + 3 \, x^{\frac{1}{3}} - 12 \, x^{\frac{1}{12}} - 2 \, \log \left (x^{\frac{1}{6}} - x^{\frac{1}{12}} + 1\right ) + 4 \, \log \left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(1/3)/(sqrt(x) + x^(1/4)),x, algorithm="fricas")

[Out]

4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 6/5*x^(5/6) - 12/7*x^(7/12) + 3
*x^(1/3) - 12*x^(1/12) - 2*log(x^(1/6) - x^(1/12) + 1) + 4*log(x^(1/12) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt [3]{x}}{\sqrt [4]{x} + \sqrt{x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/3)/(x**(1/4)+x**(1/2)),x)

[Out]

Integral(x**(1/3)/(x**(1/4) + sqrt(x)), x)

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GIAC/XCAS [A]  time = 0.285673, size = 81, normalized size = 1.07 \[ 4 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{12}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - \frac{12}{7} \, x^{\frac{7}{12}} + 3 \, x^{\frac{1}{3}} - 12 \, x^{\frac{1}{12}} - 2 \,{\rm ln}\left (x^{\frac{1}{6}} - x^{\frac{1}{12}} + 1\right ) + 4 \,{\rm ln}\left (x^{\frac{1}{12}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^(1/3)/(sqrt(x) + x^(1/4)),x, algorithm="giac")

[Out]

4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^(1/12) - 1)) + 6/5*x^(5/6) - 12/7*x^(7/12) + 3
*x^(1/3) - 12*x^(1/12) - 2*ln(x^(1/6) - x^(1/12) + 1) + 4*ln(x^(1/12) + 1)