Optimal. Leaf size=38 \[ \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-\sqrt{a+\frac{b}{x^2}} \]
[Out]
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Rubi [A] time = 0.0765745, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^2}}}{\sqrt{a}}\right )-\sqrt{a+\frac{b}{x^2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x^2]/x,x]
[Out]
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Rubi in Sympy [A] time = 6.66849, size = 31, normalized size = 0.82 \[ \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{2}}}}{\sqrt{a}} \right )} - \sqrt{a + \frac{b}{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)**(1/2)/x,x)
[Out]
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Mathematica [A] time = 0.0406865, size = 54, normalized size = 1.42 \[ \sqrt{a+\frac{b}{x^2}} \left (\frac{\sqrt{a} x \log \left (\sqrt{a} \sqrt{a x^2+b}+a x\right )}{\sqrt{a x^2+b}}-1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x^2]/x,x]
[Out]
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Maple [B] time = 0.01, size = 81, normalized size = 2.1 \[{\frac{1}{b}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ({a}^{{\frac{3}{2}}}\sqrt{a{x}^{2}+b}{x}^{2}- \left ( a{x}^{2}+b \right ) ^{{\frac{3}{2}}}\sqrt{a}+\ln \left ( \sqrt{a}x+\sqrt{a{x}^{2}+b} \right ) xab \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}{\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)^(1/2)/x,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(sqrt(a + b/x^2)/x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.248816, size = 1, normalized size = 0.03 \[ \left [\frac{1}{2} \, \sqrt{a} \log \left (-2 \, a x^{2} - 2 \, \sqrt{a} x^{2} \sqrt{\frac{a x^{2} + b}{x^{2}}} - b\right ) - \sqrt{\frac{a x^{2} + b}{x^{2}}}, \sqrt{-a} \arctan \left (\frac{a}{\sqrt{-a} \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) - \sqrt{\frac{a x^{2} + b}{x^{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^2)/x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.23451, size = 56, normalized size = 1.47 \[ \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a} x}{\sqrt{b}} \right )} - \frac{a x}{\sqrt{b} \sqrt{\frac{a x^{2}}{b} + 1}} - \frac{\sqrt{b}}{x \sqrt{\frac{a x^{2}}{b} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)**(1/2)/x,x)
[Out]
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GIAC/XCAS [A] time = 0.247001, size = 82, normalized size = 2.16 \[ -\frac{1}{2} \, \sqrt{a}{\rm ln}\left ({\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \, \sqrt{a} b{\rm sign}\left (x\right )}{{\left (\sqrt{a} x - \sqrt{a x^{2} + b}\right )}^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^2)/x,x, algorithm="giac")
[Out]