Optimal. Leaf size=38 \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{2}{a \left (a+b \sqrt{x}\right )} \]
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Rubi [A] time = 0.0219209, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 44} \[ -\frac{2 \log \left (a+b \sqrt{x}\right )}{a^2}+\frac{\log (x)}{a^2}+\frac{2}{a \left (a+b \sqrt{x}\right )} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^2 x} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x (a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x}-\frac{b}{a (a+b x)^2}-\frac{b}{a^2 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2}{a \left (a+b \sqrt{x}\right )}-\frac{2 \log \left (a+b \sqrt{x}\right )}{a^2}+\frac{\log (x)}{a^2}\\ \end{align*}
Mathematica [A] time = 0.0216941, size = 33, normalized size = 0.87 \[ \frac{\frac{2 a}{a+b \sqrt{x}}-2 \log \left (a+b \sqrt{x}\right )+\log (x)}{a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 35, normalized size = 0.9 \begin{align*}{\frac{\ln \left ( x \right ) }{{a}^{2}}}-2\,{\frac{\ln \left ( a+b\sqrt{x} \right ) }{{a}^{2}}}+2\,{\frac{1}{a \left ( a+b\sqrt{x} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.966741, size = 46, normalized size = 1.21 \begin{align*} \frac{2}{a b \sqrt{x} + a^{2}} - \frac{2 \, \log \left (b \sqrt{x} + a\right )}{a^{2}} + \frac{\log \left (x\right )}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.26779, size = 142, normalized size = 3.74 \begin{align*} \frac{2 \,{\left (a b \sqrt{x} - a^{2} -{\left (b^{2} x - a^{2}\right )} \log \left (b \sqrt{x} + a\right ) +{\left (b^{2} x - a^{2}\right )} \log \left (\sqrt{x}\right )\right )}}{a^{2} b^{2} x - a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.11896, size = 148, normalized size = 3.89 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x} & \text{for}\: a = 0 \wedge b = 0 \\\frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \\- \frac{1}{b^{2} x} & \text{for}\: a = 0 \\\frac{a \sqrt{x} \log{\left (x \right )}}{a^{3} \sqrt{x} + a^{2} b x} - \frac{2 a \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{3} \sqrt{x} + a^{2} b x} + \frac{b x \log{\left (x \right )}}{a^{3} \sqrt{x} + a^{2} b x} - \frac{2 b x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{3} \sqrt{x} + a^{2} b x} - \frac{2 b x}{a^{3} \sqrt{x} + a^{2} b x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12829, size = 49, normalized size = 1.29 \begin{align*} -\frac{2 \, \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{2}} + \frac{\log \left ({\left | x \right |}\right )}{a^{2}} + \frac{2}{{\left (b \sqrt{x} + a\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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