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 = 0.0091657, 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, Rules used = {266, 36, 29, 31} \[ \frac{\log (x)}{a}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right ) x} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x (a+b x)} \, dx,x,\sqrt{x}\right )\\ &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sqrt{x}\right )}{a}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{a+b x} \, dx,x,\sqrt{x}\right )}{a}\\ &=-\frac{2 \log \left (a+b \sqrt{x}\right )}{a}+\frac{\log (x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0035301, size = 22, normalized size = 1. \[ \frac{\log (x)}{a}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 21, normalized size = 1. \begin{align*}{\frac{\ln \left ( x \right ) }{a}}-2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967363, size = 27, normalized size = 1.23 \begin{align*} -\frac{2 \, \log \left (b \sqrt{x} + a\right )}{a} + \frac{\log \left (x\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.30794, size = 57, normalized size = 2.59 \begin{align*} -\frac{2 \,{\left (\log \left (b \sqrt{x} + a\right ) - \log \left (\sqrt{x}\right )\right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.354577, size = 37, normalized size = 1.68 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\sqrt{x}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{b \sqrt{x}} & \text{for}\: a = 0 \\\frac{\log{\left (x \right )}}{a} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{a} - \frac{2 \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09593, size = 30, normalized size = 1.36 \begin{align*} -\frac{2 \, \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a} + \frac{\log \left ({\left | x \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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