3.774 \(\int \frac{1+\sqrt{x}}{x^{5/6}+x^{7/6}} \, dx\)

Optimal. Leaf size=26 \[ 3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right )+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0605814, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ 3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right )+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + Sqrt[x])/(x^(5/6) + x^(7/6)),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - 3 \log{\left (\sqrt [3]{x} + 1 \right )} + 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} + 6 \int ^{\sqrt [6]{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x**(1/2))/(x**(5/6)+x**(7/6)),x)

[Out]

-3*log(x**(1/3) + 1) + 6*atan(x**(1/6)) + 6*Integral(x, (x, x**(1/6)))

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Mathematica [A]  time = 0.0154289, size = 26, normalized size = 1. \[ 3 \sqrt [3]{x}-3 \log \left (\sqrt [3]{x}+1\right )+6 \tan ^{-1}\left (\sqrt [6]{x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + Sqrt[x])/(x^(5/6) + x^(7/6)),x]

[Out]

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

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Maple [A]  time = 0.005, size = 21, normalized size = 0.8 \[ 3\,\sqrt [3]{x}+6\,\arctan \left ( \sqrt [6]{x} \right ) -3\,\ln \left ( 1+\sqrt [3]{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x^(1/2))/(x^(5/6)+x^(7/6)),x)

[Out]

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

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Maxima [A]  time = 0.755672, size = 27, normalized size = 1.04 \[ 3 \, x^{\frac{1}{3}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x) + 1)/(x^(7/6) + x^(5/6)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.278808, size = 27, normalized size = 1.04 \[ 3 \, x^{\frac{1}{3}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) - 3 \, \log \left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x) + 1)/(x^(7/6) + x^(5/6)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 26.8895, size = 24, normalized size = 0.92 \[ 3 \sqrt [3]{x} - 3 \log{\left (\sqrt [3]{x} + 1 \right )} + 6 \operatorname{atan}{\left (\sqrt [6]{x} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x**(1/2))/(x**(5/6)+x**(7/6)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.262336, size = 27, normalized size = 1.04 \[ 3 \, x^{\frac{1}{3}} + 6 \, \arctan \left (x^{\frac{1}{6}}\right ) - 3 \,{\rm ln}\left (x^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sqrt(x) + 1)/(x^(7/6) + x^(5/6)),x, algorithm="giac")

[Out]

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