Optimal. Leaf size=123 \[ x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} \sqrt{c}}-2 \sqrt{b} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b} \sqrt{c+\frac{d}{x}}}\right ) \]
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Rubi [A] time = 0.353574, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261 \[ x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+\frac{(a d+b c) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+\frac{b}{x}}}{\sqrt{a} \sqrt{c+\frac{d}{x}}}\right )}{\sqrt{a} \sqrt{c}}-2 \sqrt{b} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+\frac{b}{x}}}{\sqrt{b} \sqrt{c+\frac{d}{x}}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x]*Sqrt[c + d/x],x]
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Rubi in Sympy [A] time = 32.876, size = 104, normalized size = 0.85 \[ - 2 \sqrt{b} \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{b} \sqrt{c + \frac{d}{x}}}{\sqrt{d} \sqrt{a + \frac{b}{x}}} \right )} + x \sqrt{a + \frac{b}{x}} \sqrt{c + \frac{d}{x}} + \frac{\left (a d + b c\right ) \operatorname{atanh}{\left (\frac{\sqrt{c} \sqrt{a + \frac{b}{x}}}{\sqrt{a} \sqrt{c + \frac{d}{x}}} \right )}}{\sqrt{a} \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d/x)**(1/2)*(a+b/x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.355393, size = 168, normalized size = 1.37 \[ x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}-\sqrt{b} \sqrt{d} \log \left (2 \sqrt{b} \sqrt{d} x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+a d x+b c x+2 b d\right )+\frac{(a d+b c) \log \left (2 \sqrt{a} \sqrt{c} x \sqrt{a+\frac{b}{x}} \sqrt{c+\frac{d}{x}}+a (2 c x+d)+b c\right )}{2 \sqrt{a} \sqrt{c}}+\sqrt{b} \sqrt{d} \log (x) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x]*Sqrt[c + d/x],x]
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Maple [B] time = 0.033, size = 253, normalized size = 2.1 \[{\frac{x}{2}\sqrt{{\frac{cx+d}{x}}}\sqrt{{\frac{ax+b}{x}}} \left ( \sqrt{bd}\ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ac{x}^{2}+adx+bcx+bd}\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) ad+\sqrt{bd}\ln \left ({\frac{1}{2} \left ( 2\,acx+2\,\sqrt{ac{x}^{2}+adx+bcx+bd}\sqrt{ac}+ad+bc \right ){\frac{1}{\sqrt{ac}}}} \right ) bc-2\,bd\ln \left ({\frac{adx+bcx+2\,\sqrt{bd}\sqrt{ac{x}^{2}+adx+bcx+bd}+2\,bd}{x}} \right ) \sqrt{ac}+2\,\sqrt{ac{x}^{2}+adx+bcx+bd}\sqrt{ac}\sqrt{bd} \right ){\frac{1}{\sqrt{ac{x}^{2}+adx+bcx+bd}}}{\frac{1}{\sqrt{ac}}}{\frac{1}{\sqrt{bd}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d/x)^(1/2)*(a+b/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 = 0.728648, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
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. \[ \int \sqrt{a + \frac{b}{x}} \sqrt{c + \frac{d}{x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c+d/x)**(1/2)*(a+b/x)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + \frac{b}{x}} \sqrt{c + \frac{d}{x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x)*sqrt(c + d/x),x, algorithm="giac")
[Out]