\(\int \frac {2 \sqrt {-1+x}+x}{\sqrt {-1+x} x} \, dx\) [808]

Optimal result

Integrand size = 22, antiderivative size = 14 \[ \int \frac {2 \sqrt {-1+x}+x}{\sqrt {-1+x} x} \, dx=2 \sqrt {-1+x}+2 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {6820} \[ \int \frac {2 \sqrt {-1+x}+x}{\sqrt {-1+x} x} \, dx=2 \sqrt {x-1}+2 \log (x) \]

[In]

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

[Out]

2*Sqrt[-1 + x] + 2*Log[x]

Rule 6820

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {2 \sqrt {-1+x}+x}{\sqrt {-1+x} x} \, dx=2 \sqrt {-1+x}+2 \log (x) \]

[In]

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

[Out]

2*Sqrt[-1 + x] + 2*Log[x]

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(2 \ln \left (x \right )+2 \sqrt {x -1}\)	\(13\)
default	\(2 \ln \left (x \right )+2 \sqrt {x -1}\)	\(13\)
trager	\(2 \sqrt {x -1}-2 \ln \left (\frac {1}{x}\right )\)	\(15\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2 \sqrt {-1+x}+x}{\sqrt {-1+x} x} \, dx=2 \, \sqrt {x - 1} + 2 \, \log \left (x\right ) \]

[In]

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

[Out]

2*sqrt(x - 1) + 2*log(x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2 \sqrt {-1+x}+x}{\sqrt {-1+x} x} \, dx=2 \sqrt {x - 1} + 2 \log {\left (x \right )} \]

[In]

integrate((x+2*(-1+x)**(1/2))/x/(-1+x)**(1/2),x)

[Out]

2*sqrt(x - 1) + 2*log(x)

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2 \sqrt {-1+x}+x}{\sqrt {-1+x} x} \, dx=2 \, \sqrt {x - 1} + 2 \, \log \left (x\right ) \]

[In]

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

[Out]

2*sqrt(x - 1) + 2*log(x)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2 \sqrt {-1+x}+x}{\sqrt {-1+x} x} \, dx=2 \, \sqrt {x - 1} + 2 \, \log \left (x\right ) \]

[In]

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

[Out]

2*sqrt(x - 1) + 2*log(x)

Mupad [B] (verification not implemented)

Time = 20.38 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {2 \sqrt {-1+x}+x}{\sqrt {-1+x} x} \, dx=2\,\ln \left (x\right )+2\,\sqrt {x-1} \]

[In]

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

[Out]

2*log(x) + 2*(x - 1)^(1/2)