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 = 0.059137, 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 \[ \frac{x \sqrt{a+\frac{b}{x}}}{a}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x}}}{\sqrt{a}}\right )}{a^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/Sqrt[a + b/x],x]
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Rubi in Sympy [A] time = 5.35715, size = 32, normalized size = 0.74 \[ \frac{x \sqrt{a + \frac{b}{x}}}{a} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x}}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0467834, size = 53, normalized size = 1.23 \[ \frac{x \sqrt{a+\frac{b}{x}}}{a}-\frac{b \log \left (2 \sqrt{a} x \sqrt{a+\frac{b}{x}}+2 a x+b\right )}{2 a^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/Sqrt[a + b/x],x]
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Maple [A] time = 0.005, size = 70, normalized size = 1.6 \[ -{\frac{x}{2}\sqrt{{\frac{ax+b}{x}}} \left ( b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) -2\,\sqrt{x \left ( ax+b \right ) }\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( ax+b \right ) }}}{a}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x)^(1/2),x)
[Out]
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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(1/sqrt(a + b/x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247426, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{a} x \sqrt{\frac{a x + b}{x}} + b \log \left (-2 \, a x \sqrt{\frac{a x + b}{x}} +{\left (2 \, a x + b\right )} \sqrt{a}\right )}{2 \, a^{\frac{3}{2}}}, \frac{\sqrt{-a} x \sqrt{\frac{a x + b}{x}} + b \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x + b}{x}}}\right )}{\sqrt{-a} a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a + b/x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 7.83873, size = 44, normalized size = 1.02 \[ \frac{\sqrt{b} \sqrt{x} \sqrt{\frac{a x}{b} + 1}}{a} - \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{b}} \right )}}{a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.237001, size = 96, normalized size = 2.23 \[ -\frac{b{\rm ln}\left ({\left | b \right |}\right ){\rm sign}\left (x\right )}{2 \, a^{\frac{3}{2}}} + \frac{b{\rm ln}\left ({\left | -2 \,{\left (\sqrt{a} x - \sqrt{a x^{2} + b x}\right )} \sqrt{a} - b \right |}\right )}{2 \, a^{\frac{3}{2}}{\rm sign}\left (x\right )} + \frac{\sqrt{a x^{2} + b x}}{a{\rm sign}\left (x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/sqrt(a + b/x),x, algorithm="giac")
[Out]