3.232 \(\int \frac{1+\sqrt{x}}{-1+\sqrt{x}} \, dx\)

Optimal. Leaf size=21 \[ x+4 \sqrt{x}+4 \log \left (1-\sqrt{x}\right ) \]

[Out]

4*Sqrt[x] + x + 4*Log[1 - Sqrt[x]]

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Rubi [A]  time = 0.0281364, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ x+4 \sqrt{x}+4 \log \left (1-\sqrt{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

4*Sqrt[x] + x + 4*Log[1 - Sqrt[x]]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 4 \sqrt{x} + 4 \log{\left (- \sqrt{x} + 1 \right )} + 2 \int ^{\sqrt{x}} x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(x) + 4*log(-sqrt(x) + 1) + 2*Integral(x, (x, sqrt(x)))

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Mathematica [A]  time = 0.00737561, size = 20, normalized size = 0.95 \[ x+4 \sqrt{x}+4 \log \left (\sqrt{x}-1\right )-5 \]

Antiderivative was successfully verified.

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

[Out]

-5 + 4*Sqrt[x] + x + 4*Log[-1 + Sqrt[x]]

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \[ x+4\,\sqrt{x}+4\,\ln \left ( \sqrt{x}-1 \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x^(1/2))/(x^(1/2)-1),x)

[Out]

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

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Maxima [A]  time = 1.36087, size = 20, normalized size = 0.95 \[ x + 4 \, \sqrt{x} + 4 \, \log \left (\sqrt{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + 4*sqrt(x) + 4*log(sqrt(x) - 1)

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Fricas [A]  time = 0.20237, size = 20, normalized size = 0.95 \[ x + 4 \, \sqrt{x} + 4 \, \log \left (\sqrt{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + 4*sqrt(x) + 4*log(sqrt(x) - 1)

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Sympy [A]  time = 0.162879, size = 17, normalized size = 0.81 \[ 4 \sqrt{x} + x + 4 \log{\left (\sqrt{x} - 1 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*sqrt(x) + x + 4*log(sqrt(x) - 1)

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GIAC/XCAS [A]  time = 0.208148, size = 22, normalized size = 1.05 \[ x + 4 \, \sqrt{x} + 4 \,{\rm ln}\left ({\left | \sqrt{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x + 4*sqrt(x) + 4*ln(abs(sqrt(x) - 1))