Optimal. Leaf size=39 \[ 2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 \sqrt{a+\frac{b}{x}} \]
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Rubi [A] time = 0.0192805, antiderivative size = 39, 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, 50, 63, 208} \[ 2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 \sqrt{a+\frac{b}{x}} \]
Antiderivative was successfully verified.
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Rule 266
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+\frac{b}{x}}}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x} \, dx,x,\frac{1}{x}\right )\\ &=-2 \sqrt{a+\frac{b}{x}}-a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=-2 \sqrt{a+\frac{b}{x}}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{b}\\ &=-2 \sqrt{a+\frac{b}{x}}+2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )\\ \end{align*}
Mathematica [A] time = 0.0103471, size = 39, normalized size = 1. \[ 2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )-2 \sqrt{a+\frac{b}{x}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 103, normalized size = 2.6 \begin{align*} -{\frac{1}{bx}\sqrt{{\frac{ax+b}{x}}} \left ( -2\,\sqrt{a{x}^{2}+bx}{a}^{3/2}{x}^{2}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ){x}^{2}ab+2\, \left ( a{x}^{2}+bx \right ) ^{3/2}\sqrt{a} \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{\frac{1}{\sqrt{a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6992, size = 208, normalized size = 5.33 \begin{align*} \left [\sqrt{a} \log \left (2 \, a x + 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right ) - 2 \, \sqrt{\frac{a x + b}{x}}, -2 \, \sqrt{-a} \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right ) - 2 \, \sqrt{\frac{a x + b}{x}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.84498, size = 68, normalized size = 1.74 \begin{align*} 2 \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )} - \frac{2 a \sqrt{x}}{\sqrt{b} \sqrt{\frac{a x}{b} + 1}} - \frac{2 \sqrt{b}}{\sqrt{x} \sqrt{\frac{a x}{b} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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