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