Optimal. Leaf size=47 \[ -\frac{2 b^2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{b^2 \log (x)}{a^3}+\frac{2 b}{a^2 \sqrt{x}}-\frac{1}{a x} \]
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Rubi [A] time = 0.0288228, antiderivative size = 47, 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} \[ -\frac{2 b^2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{b^2 \log (x)}{a^3}+\frac{2 b}{a^2 \sqrt{x}}-\frac{1}{a x} \]
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 ) x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a x^3}-\frac{b}{a^2 x^2}+\frac{b^2}{a^3 x}-\frac{b^3}{a^3 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=-\frac{1}{a x}+\frac{2 b}{a^2 \sqrt{x}}-\frac{2 b^2 \log \left (a+b \sqrt{x}\right )}{a^3}+\frac{b^2 \log (x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0231124, size = 44, normalized size = 0.94 \[ \frac{-2 b^2 x \log \left (a+b \sqrt{x}\right )-a \left (a-2 b \sqrt{x}\right )+b^2 x \log (x)}{a^3 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 44, normalized size = 0.9 \begin{align*} -{\frac{1}{ax}}+{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{3}}}-2\,{\frac{{b}^{2}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{3}}}+2\,{\frac{b}{{a}^{2}\sqrt{x}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.955656, size = 58, normalized size = 1.23 \begin{align*} -\frac{2 \, b^{2} \log \left (b \sqrt{x} + a\right )}{a^{3}} + \frac{b^{2} \log \left (x\right )}{a^{3}} + \frac{2 \, b \sqrt{x} - a}{a^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.33225, size = 113, normalized size = 2.4 \begin{align*} -\frac{2 \, b^{2} x \log \left (b \sqrt{x} + a\right ) - 2 \, b^{2} x \log \left (\sqrt{x}\right ) - 2 \, a b \sqrt{x} + a^{2}}{a^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.25754, size = 68, normalized size = 1.45 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{3 b x^{\frac{3}{2}}} & \text{for}\: a = 0 \\- \frac{1}{a x} & \text{for}\: b = 0 \\- \frac{1}{a x} + \frac{2 b}{a^{2} \sqrt{x}} + \frac{b^{2} \log{\left (x \right )}}{a^{3}} - \frac{2 b^{2} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09667, size = 65, normalized size = 1.38 \begin{align*} -\frac{2 \, b^{2} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{3}} + \frac{b^{2} \log \left ({\left | x \right |}\right )}{a^{3}} + \frac{2 \, a b \sqrt{x} - a^{2}}{a^{3} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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