Optimal. Leaf size=43 \[ \frac{x \sqrt{a+\frac{b}{x}}}{a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]
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Rubi [A] time = 0.0171796, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {242, 51, 63, 208} \[ \frac{x \sqrt{a+\frac{b}{x}}}{a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Rule 242
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+\frac{b}{x}}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{a+\frac{b}{x}} x}{a}+\frac{b \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\frac{1}{x}\right )}{2 a}\\ &=\frac{\sqrt{a+\frac{b}{x}} x}{a}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+\frac{b}{x}}\right )}{a}\\ &=\frac{\sqrt{a+\frac{b}{x}} x}{a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.018456, size = 43, normalized size = 1. \[ \frac{x \sqrt{a+\frac{b}{x}}}{a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 71, normalized size = 1.7 \begin{align*}{\frac{x}{2}\sqrt{{\frac{ax+b}{x}}} \left ( 2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}-b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+b \right ) x}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) \right ){\frac{1}{\sqrt{ \left ( ax+b \right ) x}}}{a}^{-{\frac{3}{2}}}} \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 = 1.79034, size = 239, normalized size = 5.56 \begin{align*} \left [\frac{2 \, a x \sqrt{\frac{a x + b}{x}} + \sqrt{a} b \log \left (2 \, a x - 2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b\right )}{2 \, a^{2}}, \frac{a x \sqrt{\frac{a x + b}{x}} + \sqrt{-a} b \arctan \left (\frac{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}{a}\right )}{a^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.09912, size = 44, normalized size = 1.02 \begin{align*} \frac{\sqrt{b} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{a} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15519, size = 96, normalized size = 2.23 \begin{align*} -\frac{b \log \left ({\left | b \right |}\right ) \mathrm{sgn}\left (x\right )}{2 \, a^{\frac{3}{2}}} + \frac{b \log \left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right )}{2 \, a^{\frac{3}{2}} \mathrm{sgn}\left (x\right )} + \frac{\sqrt{a x^{2} + b x}}{a \mathrm{sgn}\left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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