3.2441 \(\int \frac{1}{1+\sqrt [5]{x}} \, dx\)

Optimal. Leaf size=45 \[ \frac{5 x^{4/5}}{4}-\frac{5 x^{3/5}}{3}+\frac{5 x^{2/5}}{2}-5 \sqrt [5]{x}+5 \log \left (\sqrt [5]{x}+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.04009, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{5 x^{4/5}}{4}-\frac{5 x^{3/5}}{3}+\frac{5 x^{2/5}}{2}-5 \sqrt [5]{x}+5 \log \left (\sqrt [5]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 + x^(1/5))^(-1),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{5 x^{\frac{4}{5}}}{4} - \frac{5 x^{\frac{3}{5}}}{3} - 5 \sqrt [5]{x} + 5 \log{\left (\sqrt [5]{x} + 1 \right )} + 5 \int ^{\sqrt [5]{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*x**(4/5)/4 - 5*x**(3/5)/3 - 5*x**(1/5) + 5*log(x**(1/5) + 1) + 5*Integral(x, (
x, x**(1/5)))

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Mathematica [A]  time = 0.0160331, size = 45, normalized size = 1. \[ \frac{5 x^{4/5}}{4}-\frac{5 x^{3/5}}{3}+\frac{5 x^{2/5}}{2}-5 \sqrt [5]{x}+5 \log \left (\sqrt [5]{x}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x^(1/5))^(-1),x]

[Out]

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

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Maple [B]  time = 0.065, size = 79, normalized size = 1.8 \[ \ln \left ( 1+x \right ) +{\frac{5}{2}{x}^{{\frac{2}{5}}}}+4\,\ln \left ( 1+\sqrt [5]{x} \right ) -\ln \left ( -\sqrt [5]{x}\sqrt{5}+2\,{x}^{2/5}-\sqrt [5]{x}+2 \right ) -\ln \left ( \sqrt [5]{x}\sqrt{5}+2\,{x}^{2/5}-\sqrt [5]{x}+2 \right ) +{\frac{5}{4}{x}^{{\frac{4}{5}}}}-5\,\sqrt [5]{x}-{\frac{5}{3}{x}^{{\frac{3}{5}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+x^(1/5)),x)

[Out]

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

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Maxima [A]  time = 1.44003, size = 57, normalized size = 1.27 \[ \frac{5}{4} \,{\left (x^{\frac{1}{5}} + 1\right )}^{4} - \frac{20}{3} \,{\left (x^{\frac{1}{5}} + 1\right )}^{3} + 15 \,{\left (x^{\frac{1}{5}} + 1\right )}^{2} - 20 \, x^{\frac{1}{5}} + 5 \, \log \left (x^{\frac{1}{5}} + 1\right ) - 20 \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/4*(x^(1/5) + 1)^4 - 20/3*(x^(1/5) + 1)^3 + 15*(x^(1/5) + 1)^2 - 20*x^(1/5) + 5
*log(x^(1/5) + 1) - 20

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Fricas [A]  time = 0.237965, size = 39, normalized size = 0.87 \[ \frac{5}{4} \, x^{\frac{4}{5}} - \frac{5}{3} \, x^{\frac{3}{5}} + \frac{5}{2} \, x^{\frac{2}{5}} - 5 \, x^{\frac{1}{5}} + 5 \, \log \left (x^{\frac{1}{5}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 12.8416, size = 41, normalized size = 0.91 \[ \frac{5 x^{\frac{4}{5}}}{4} - \frac{5 x^{\frac{3}{5}}}{3} + \frac{5 x^{\frac{2}{5}}}{2} - 5 \sqrt [5]{x} + 5 \log{\left (\sqrt [5]{x} + 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+x**(1/5)),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.214154, size = 39, normalized size = 0.87 \[ \frac{5}{4} \, x^{\frac{4}{5}} - \frac{5}{3} \, x^{\frac{3}{5}} + \frac{5}{2} \, x^{\frac{2}{5}} - 5 \, x^{\frac{1}{5}} + 5 \,{\rm ln}\left (x^{\frac{1}{5}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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