Optimal. Leaf size=81 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{3/2}}+\frac{x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}}{c} \]
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Rubi [A] time = 0.0520053, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {375, 94, 93, 208} \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{3/2}}+\frac{x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}}{c} \]
Antiderivative was successfully verified.
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Rule 375
Rule 94
Rule 93
Rule 208
Rubi steps
\begin{align*} \int \frac{\sqrt{a+\frac{b}{x}}}{\sqrt{c+\frac{d}{x}}} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{a+b x}}{x^2 \sqrt{c+d x}} \, dx,x,\frac{1}{x}\right )\\ &=\frac{\sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}} x}{c}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x} \sqrt{c+d x}} \, dx,x,\frac{1}{x}\right )}{2 c}\\ &=\frac{\sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}} x}{c}-\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{-a+c x^2} \, dx,x,\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{c+\frac{d}{x}}}\right )}{c}\\ &=\frac{\sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}} x}{c}+\frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0607665, size = 81, normalized size = 1. \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{3/2}}+\frac{x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}}{c} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.024, size = 155, normalized size = 1.9 \begin{align*} -{\frac{x}{2\,c}\sqrt{{\frac{ax+b}{x}}}\sqrt{{\frac{cx+d}{x}}} \left ( \ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) ad-\ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) bc-2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }}}{\frac{1}{\sqrt{ac}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{b}{x}}}{\sqrt{c + \frac{d}{x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21108, size = 551, normalized size = 6.8 \begin{align*} \left [\frac{4 \, a c x \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} - \sqrt{a c}{\left (b c - a d\right )} \log \left (-8 \, a^{2} c^{2} x^{2} - b^{2} c^{2} - 6 \, a b c d - a^{2} d^{2} + 4 \,{\left (2 \, a c x^{2} +{\left (b c + a d\right )} x\right )} \sqrt{a c} \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} - 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )}{4 \, a c^{2}}, \frac{2 \, a c x \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} - \sqrt{-a c}{\left (b c - a d\right )} \arctan \left (\frac{2 \, \sqrt{-a c} x \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}}}{2 \, a c x + b c + a d}\right )}{2 \, a c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{b}{x}}}{\sqrt{c + \frac{d}{x}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + \frac{b}{x}}}{\sqrt{c + \frac{d}{x}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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