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

Optimal. Leaf size=74 \[ \frac{6 x^{5/6}}{5}+2 \sqrt{x}-3 \sqrt [3]{x}-4 \log \left (\sqrt [6]{x}+1\right )-\log \left (\sqrt [3]{x}-\sqrt [6]{x}+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [6]{x}}{\sqrt{3}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.182749, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471 \[ \frac{6 x^{5/6}}{5}+2 \sqrt{x}-3 \sqrt [3]{x}-4 \log \left (\sqrt [6]{x}+1\right )-\log \left (\sqrt [3]{x}-\sqrt [6]{x}+1\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-2 \sqrt [6]{x}}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.92872, size = 65, normalized size = 0.88 \[ \frac{6 x^{\frac{5}{6}}}{5} - 3 \sqrt [3]{x} + 2 \sqrt{x} - 3 \log{\left (\sqrt [6]{x} + 1 \right )} - \log{\left (\sqrt{x} + 1 \right )} + 2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [6]{x}}{3} - \frac{1}{3}\right ) \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0294826, size = 82, normalized size = 1.11 \[ \frac{6 x^{5/6}}{5}+2 \sqrt{x}-3 \sqrt [3]{x}-2 \log \left (\sqrt [6]{x}+1\right )+\log \left (\sqrt [3]{x}-\sqrt [6]{x}+1\right )-2 \log \left (\sqrt{x}+1\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [6]{x}-1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 56, normalized size = 0.8 \[{\frac{6}{5}{x}^{{\frac{5}{6}}}}+2\,\sqrt{x}-3\,\sqrt [3]{x}-4\,\ln \left ( 1+\sqrt [6]{x} \right ) -\ln \left ( 1-\sqrt [6]{x}+\sqrt [3]{x} \right ) +2\,\sqrt{3}\arctan \left ( 1/3\, \left ( 2\,\sqrt [6]{x}-1 \right ) \sqrt{3} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x^(1/3))/(1+x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 0.773578, size = 74, normalized size = 1. \[ 2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{6}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} + 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} - \log \left (x^{\frac{1}{3}} - x^{\frac{1}{6}} + 1\right ) - 4 \, \log \left (x^{\frac{1}{6}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.273009, size = 74, normalized size = 1. \[ 2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{6}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} + 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} - \log \left (x^{\frac{1}{3}} - x^{\frac{1}{6}} + 1\right ) - 4 \, \log \left (x^{\frac{1}{6}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.83023, size = 155, normalized size = 2.09 \[ \frac{16 x^{\frac{5}{6}} \Gamma \left (\frac{8}{3}\right )}{5 \Gamma \left (\frac{11}{3}\right )} - \frac{8 \sqrt [3]{x} \Gamma \left (\frac{8}{3}\right )}{\Gamma \left (\frac{11}{3}\right )} + 2 \sqrt{x} - 2 \log{\left (\sqrt{x} + 1 \right )} - \frac{16 e^{\frac{10 i \pi }{3}} \log{\left (- \sqrt [6]{x} e^{\frac{i \pi }{3}} + 1 \right )} \Gamma \left (\frac{8}{3}\right )}{3 \Gamma \left (\frac{11}{3}\right )} - \frac{16 \log{\left (- \sqrt [6]{x} e^{i \pi } + 1 \right )} \Gamma \left (\frac{8}{3}\right )}{3 \Gamma \left (\frac{11}{3}\right )} - \frac{16 e^{\frac{2 i \pi }{3}} \log{\left (- \sqrt [6]{x} e^{\frac{5 i \pi }{3}} + 1 \right )} \Gamma \left (\frac{8}{3}\right )}{3 \Gamma \left (\frac{11}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16*x**(5/6)*gamma(8/3)/(5*gamma(11/3)) - 8*x**(1/3)*gamma(8/3)/gamma(11/3) + 2*s
qrt(x) - 2*log(sqrt(x) + 1) - 16*exp(10*I*pi/3)*log(-x**(1/6)*exp_polar(I*pi/3)
+ 1)*gamma(8/3)/(3*gamma(11/3)) - 16*log(-x**(1/6)*exp_polar(I*pi) + 1)*gamma(8/
3)/(3*gamma(11/3)) - 16*exp(2*I*pi/3)*log(-x**(1/6)*exp_polar(5*I*pi/3) + 1)*gam
ma(8/3)/(3*gamma(11/3))

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GIAC/XCAS [A]  time = 0.276657, size = 74, normalized size = 1. \[ 2 \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{\frac{1}{6}} - 1\right )}\right ) + \frac{6}{5} \, x^{\frac{5}{6}} + 2 \, \sqrt{x} - 3 \, x^{\frac{1}{3}} -{\rm ln}\left (x^{\frac{1}{3}} - x^{\frac{1}{6}} + 1\right ) - 4 \,{\rm ln}\left (x^{\frac{1}{6}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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