3.2382 \(\int \frac{\sqrt{x}}{1+x^{2/3}} \, dx\)

Optimal. Leaf size=117 \[ \frac{6 x^{5/6}}{5}-6 \sqrt [6]{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]

[Out]

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

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Rubi [A]  time = 0.182967, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ \frac{6 x^{5/6}}{5}-6 \sqrt [6]{x}-\frac{3 \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}+\frac{3 \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )}{2 \sqrt{2}}-\frac{3 \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )}{\sqrt{2}}+\frac{3 \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(1 + x^(2/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.8284, size = 112, normalized size = 0.96 \[ \frac{6 x^{\frac{5}{6}}}{5} - 6 \sqrt [6]{x} - \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [6]{x} + \sqrt [3]{x} + 1 \right )}}{4} + \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [6]{x} + \sqrt [3]{x} + 1 \right )}}{4} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [6]{x} - 1 \right )}}{2} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \sqrt [6]{x} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0529053, size = 115, normalized size = 0.98 \[ \frac{3}{20} \left (8 x^{5/6}-40 \sqrt [6]{x}-5 \sqrt{2} \log \left (\sqrt [3]{x}-\sqrt{2} \sqrt [6]{x}+1\right )+5 \sqrt{2} \log \left (\sqrt [3]{x}+\sqrt{2} \sqrt [6]{x}+1\right )-10 \sqrt{2} \tan ^{-1}\left (1-\sqrt{2} \sqrt [6]{x}\right )+10 \sqrt{2} \tan ^{-1}\left (\sqrt{2} \sqrt [6]{x}+1\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

(3*(-40*x^(1/6) + 8*x^(5/6) - 10*Sqrt[2]*ArcTan[1 - Sqrt[2]*x^(1/6)] + 10*Sqrt[2
]*ArcTan[1 + Sqrt[2]*x^(1/6)] - 5*Sqrt[2]*Log[1 - Sqrt[2]*x^(1/6) + x^(1/3)] + 5
*Sqrt[2]*Log[1 + Sqrt[2]*x^(1/6) + x^(1/3)]))/20

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Maple [A]  time = 0.006, size = 76, normalized size = 0.7 \[{\frac{6}{5}{x}^{{\frac{5}{6}}}}-6\,\sqrt [6]{x}+{\frac{3\,\sqrt{2}}{2}\arctan \left ( \sqrt [6]{x}\sqrt{2}-1 \right ) }+{\frac{3\,\sqrt{2}}{4}\ln \left ({1 \left ( 1+\sqrt [3]{x}+\sqrt [6]{x}\sqrt{2} \right ) \left ( 1+\sqrt [3]{x}-\sqrt [6]{x}\sqrt{2} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}}{2}\arctan \left ( 1+\sqrt [6]{x}\sqrt{2} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(1+x^(2/3)),x)

[Out]

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

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Maxima [A]  time = 1.59657, size = 119, normalized size = 1.02 \[ \frac{3}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{4} \, \sqrt{2} \log \left (\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 6 \, x^{\frac{1}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.239029, size = 163, normalized size = 1.39 \[ -3 \, \sqrt{2} \arctan \left (\frac{1}{\sqrt{2} x^{\frac{1}{6}} + \sqrt{2 \, \sqrt{2} x^{\frac{1}{6}} + 2 \, x^{\frac{1}{3}} + 2} + 1}\right ) - 3 \, \sqrt{2} \arctan \left (\frac{1}{\sqrt{2} x^{\frac{1}{6}} + \sqrt{-2 \, \sqrt{2} x^{\frac{1}{6}} + 2 \, x^{\frac{1}{3}} + 2} - 1}\right ) + \frac{3}{4} \, \sqrt{2} \log \left (2 \, \sqrt{2} x^{\frac{1}{6}} + 2 \, x^{\frac{1}{3}} + 2\right ) - \frac{3}{4} \, \sqrt{2} \log \left (-2 \, \sqrt{2} x^{\frac{1}{6}} + 2 \, x^{\frac{1}{3}} + 2\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 6 \, x^{\frac{1}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.86361, size = 187, normalized size = 1.6 \[ \frac{27 x^{\frac{5}{6}} \Gamma \left (\frac{9}{4}\right )}{10 \Gamma \left (\frac{13}{4}\right )} - \frac{27 \sqrt [6]{x} \Gamma \left (\frac{9}{4}\right )}{2 \Gamma \left (\frac{13}{4}\right )} - \frac{27 e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} + \frac{27 i e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{3 i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} + \frac{27 e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{5 i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} - \frac{27 i e^{- \frac{i \pi }{4}} \log{\left (- \sqrt [6]{x} e^{\frac{7 i \pi }{4}} + 1 \right )} \Gamma \left (\frac{9}{4}\right )}{8 \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*x**(5/6)*gamma(9/4)/(10*gamma(13/4)) - 27*x**(1/6)*gamma(9/4)/(2*gamma(13/4))
- 27*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(I*pi/4) + 1)*gamma(9/4)/(8*gamma(13/4
)) + 27*I*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(3*I*pi/4) + 1)*gamma(9/4)/(8*gamm
a(13/4)) + 27*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(5*I*pi/4) + 1)*gamma(9/4)/(8*
gamma(13/4)) - 27*I*exp(-I*pi/4)*log(-x**(1/6)*exp_polar(7*I*pi/4) + 1)*gamma(9/
4)/(8*gamma(13/4))

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GIAC/XCAS [A]  time = 0.223195, size = 119, normalized size = 1.02 \[ \frac{3}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{2} \, \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, x^{\frac{1}{6}}\right )}\right ) + \frac{3}{4} \, \sqrt{2}{\rm ln}\left (\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) - \frac{3}{4} \, \sqrt{2}{\rm ln}\left (-\sqrt{2} x^{\frac{1}{6}} + x^{\frac{1}{3}} + 1\right ) + \frac{6}{5} \, x^{\frac{5}{6}} - 6 \, x^{\frac{1}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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