3.2230 \(\int \frac{1}{\left (2+b \sqrt{x}\right ) x} \, dx\)

Optimal. Leaf size=19 \[ \frac{\log (x)}{2}-\log \left (b \sqrt{x}+2\right ) \]

[Out]

-Log[2 + b*Sqrt[x]] + Log[x]/2

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Rubi [A]  time = 0.0282215, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\log (x)}{2}-\log \left (b \sqrt{x}+2\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/((2 + b*Sqrt[x])*x),x]

[Out]

-Log[2 + b*Sqrt[x]] + Log[x]/2

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Rubi in Sympy [A]  time = 4.44836, size = 15, normalized size = 0.79 \[ \log{\left (\sqrt{x} \right )} - \log{\left (b \sqrt{x} + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sqrt(x)) - log(b*sqrt(x) + 2)

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Mathematica [A]  time = 0.00926639, size = 19, normalized size = 1. \[ \log \left (\sqrt{x}\right )-\log \left (b \sqrt{x}+2\right ) \]

Antiderivative was successfully verified.

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

[Out]

-Log[2 + b*Sqrt[x]] + Log[Sqrt[x]]

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Maple [A]  time = 0.01, size = 16, normalized size = 0.8 \[{\frac{\ln \left ( x \right ) }{2}}-\ln \left ( 2+b\sqrt{x} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(2+b*x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.43428, size = 20, normalized size = 1.05 \[ -\log \left (b \sqrt{x} + 2\right ) + \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(b*sqrt(x) + 2) + 1/2*log(x)

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Fricas [A]  time = 0.242178, size = 20, normalized size = 1.05 \[ -\log \left (b \sqrt{x} + 2\right ) + \log \left (\sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(b*sqrt(x) + 2) + log(sqrt(x))

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Sympy [A]  time = 1.78924, size = 19, normalized size = 1. \[ \begin{cases} \frac{\log{\left (x \right )}}{2} - \log{\left (\sqrt{x} + \frac{2}{b} \right )} & \text{for}\: b \neq 0 \\\frac{\log{\left (x \right )}}{2} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(2+b*x**(1/2)),x)

[Out]

Piecewise((log(x)/2 - log(sqrt(x) + 2/b), Ne(b, 0)), (log(x)/2, True))

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GIAC/XCAS [A]  time = 0.241362, size = 23, normalized size = 1.21 \[ -{\rm ln}\left ({\left | b \sqrt{x} + 2 \right |}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-ln(abs(b*sqrt(x) + 2)) + 1/2*ln(abs(x))