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 = 0.192844, 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 \[ \frac{(b c-a d) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} c^{3/2}}+\frac{x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}}{c} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x]/Sqrt[c + d/x],x]
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Rubi in Sympy [A] time = 14.8743, size = 65, normalized size = 0.8 \[ \frac{x \sqrt{a + \frac{b}{x}} \sqrt{c + \frac{d}{x}}}{c} - \frac{\left (a d - b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + \frac{b}{x}}}{\sqrt{a} \sqrt{c + \frac{d}{x}}} \right )}}{\sqrt{a} c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x)**(1/2)/(c+d/x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.19538, size = 98, normalized size = 1.21 \[ \frac{(b c-a d) \log \left (2 \sqrt{a} \sqrt{c} x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+2 a c x+a d+b c\right )}{2 \sqrt{a} c^{3/2}}+\frac{x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}}{c} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x]/Sqrt[c + d/x],x]
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Maple [B] time = 0.034, size = 155, normalized size = 1.9 \[ -{\frac{x}{2\,c}\sqrt{{\frac{ax+b}{x}}}\sqrt{{\frac{cx+d}{x}}} \left ( \ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) ad-\ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) bc-2\,\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }\sqrt{ac} \right ){\frac{1}{\sqrt{ \left ( cx+d \right ) \left ( ax+b \right ) }}}{\frac{1}{\sqrt{ac}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x)^(1/2)/(c+d/x)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)/sqrt(c + d/x),x, algorithm="maxima")
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Fricas [A] time = 0.327013, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{a c} x \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} -{\left (b c - a d\right )} \log \left (4 \,{\left (2 \, a^{2} c^{2} x^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} -{\left (8 \, a^{2} c^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x\right )} \sqrt{a c}\right )}{4 \, \sqrt{a c} c}, \frac{2 \, \sqrt{-a c} x \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}} +{\left (b c - a d\right )} \arctan \left (\frac{2 \, \sqrt{-a c} x \sqrt{\frac{a x + b}{x}} \sqrt{\frac{c x + d}{x}}}{2 \, a c x + b c + a d}\right )}{2 \, \sqrt{-a c} c}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)/sqrt(c + d/x),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + \frac{b}{x}}}{\sqrt{c + \frac{d}{x}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x)**(1/2)/(c+d/x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a + \frac{b}{x}}}{\sqrt{c + \frac{d}{x}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)/sqrt(c + d/x),x, algorithm="giac")
[Out]