3.652 \(\int \frac{2 \sqrt{-1+x}+x}{\sqrt{-1+x} x} \, dx\)

Optimal. Leaf size=14 \[ 2 \sqrt{x-1}+2 \log (x) \]

[Out]

2*Sqrt[-1 + x] + 2*Log[x]

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Rubi [A]  time = 0.193232, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ 2 \sqrt{x-1}+2 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[-1 + x] + 2*Log[x]

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Rubi in Sympy [A]  time = 11.3264, size = 12, normalized size = 0.86 \[ 2 \sqrt{x - 1} + 2 \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x - 1) + 2*log(x)

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Mathematica [A]  time = 0.00777559, size = 14, normalized size = 1. \[ 2 \sqrt{x-1}+2 \log (x) \]

Antiderivative was successfully verified.

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

[Out]

2*Sqrt[-1 + x] + 2*Log[x]

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Maple [A]  time = 0.002, size = 13, normalized size = 0.9 \[ 2\,\ln \left ( x \right ) +2\,\sqrt{-1+x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.801496, size = 16, normalized size = 1.14 \[ 2 \, \sqrt{x - 1} + 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x - 1) + 2*log(x)

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Fricas [A]  time = 0.26912, size = 16, normalized size = 1.14 \[ 2 \, \sqrt{x - 1} + 2 \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x - 1) + 2*log(x)

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Sympy [A]  time = 0.347121, size = 12, normalized size = 0.86 \[ 2 \sqrt{x - 1} + 2 \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x - 1) + 2*log(x)

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GIAC/XCAS [A]  time = 0.264567, size = 16, normalized size = 1.14 \[ 2 \, \sqrt{x - 1} + 2 \,{\rm ln}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(x - 1) + 2*ln(x)