3.2193 \(\int \frac{1}{\left (a+b \sqrt{x}\right ) x} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log (x)}{a}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a} \]

[Out]

(-2*Log[a + b*Sqrt[x]])/a + Log[x]/a

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Rubi [A]  time = 0.0300269, antiderivative size = 22, 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 \[ \frac{\log (x)}{a}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*Sqrt[x])*x),x]

[Out]

(-2*Log[a + b*Sqrt[x]])/a + Log[x]/a

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Rubi in Sympy [A]  time = 5.63783, size = 22, normalized size = 1. \[ \frac{2 \log{\left (\sqrt{x} \right )}}{a} - \frac{2 \log{\left (a + b \sqrt{x} \right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*log(sqrt(x))/a - 2*log(a + b*sqrt(x))/a

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Mathematica [A]  time = 0.00803701, size = 27, normalized size = 1.23 \[ \frac{2 \log \left (\sqrt{x}\right )}{a}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Log[a + b*Sqrt[x]])/a + (2*Log[Sqrt[x]])/a

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Maple [A]  time = 0.008, size = 21, normalized size = 1. \[{\frac{\ln \left ( x \right ) }{a}}-2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(a+b*x^(1/2)),x)

[Out]

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

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Maxima [A]  time = 1.43953, size = 27, normalized size = 1.23 \[ -\frac{2 \, \log \left (b \sqrt{x} + a\right )}{a} + \frac{\log \left (x\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*log(b*sqrt(x) + a)/a + log(x)/a

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Fricas [A]  time = 0.23689, size = 27, normalized size = 1.23 \[ -\frac{2 \,{\left (\log \left (b \sqrt{x} + a\right ) - \log \left (\sqrt{x}\right )\right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(log(b*sqrt(x) + a) - log(sqrt(x)))/a

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Sympy [A]  time = 1.16948, size = 37, normalized size = 1.68 \[ \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\- \frac{2}{b \sqrt{x}} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a} - \frac{2 \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(a+b*x**(1/2)),x)

[Out]

Piecewise((zoo/sqrt(x), Eq(a, 0) & Eq(b, 0)), (log(x)/a, Eq(b, 0)), (-2/(b*sqrt(
x)), Eq(a, 0)), (log(x)/a - 2*log(a/b + sqrt(x))/a, True))

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GIAC/XCAS [A]  time = 0.217808, size = 30, normalized size = 1.36 \[ -\frac{2 \,{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a} + \frac{{\rm ln}\left ({\left | x \right |}\right )}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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