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

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

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6688} \[ 2 \sqrt {x-1}+2 \log (x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

2*Sqrt[-1 + x] + 2*Log[x]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {align*} \int \frac {2 \sqrt {-1+x}+x}{\sqrt {-1+x} x} \, dx &=\int \left (\frac {1}{\sqrt {-1+x}}+\frac {2}{x}\right ) \, dx\\ &=2 \sqrt {-1+x}+2 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 14, normalized size = 1.00 \[ 2 \sqrt {x-1}+2 \log (x) \]

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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fricas [A]  time = 0.40, size = 12, normalized size = 0.86 \[ 2 \, \sqrt {x - 1} + 2 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+2*(-1+x)^(1/2))/x/(-1+x)^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(x - 1) + 2*log(x)

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giac [A]  time = 0.40, size = 12, normalized size = 0.86 \[ 2 \, \sqrt {x - 1} + 2 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+2*(-1+x)^(1/2))/x/(-1+x)^(1/2),x, algorithm="giac")

[Out]

2*sqrt(x - 1) + 2*log(x)

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maple [A]  time = 0.00, size = 13, normalized size = 0.93 \[ 2 \ln \relax (x )+2 \sqrt {x -1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.98, size = 12, normalized size = 0.86 \[ 2 \, \sqrt {x - 1} + 2 \, \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+2*(-1+x)^(1/2))/x/(-1+x)^(1/2),x, algorithm="maxima")

[Out]

2*sqrt(x - 1) + 2*log(x)

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mupad [B]  time = 3.39, size = 12, normalized size = 0.86 \[ 2\,\ln \relax (x)+2\,\sqrt {x-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*log(x) + 2*(x - 1)^(1/2)

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sympy [A]  time = 0.16, size = 12, normalized size = 0.86 \[ 2 \sqrt {x - 1} + 2 \log {\relax (x )} \]

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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