Optimal. Leaf size=42 \[ x \sqrt{a+\frac{b}{x^2}}-\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right ) \]
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Rubi [A] time = 0.0617865, antiderivative size = 42, 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 \[ x \sqrt{a+\frac{b}{x^2}}-\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x^2],x]
[Out]
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Rubi in Sympy [A] time = 5.38026, size = 34, normalized size = 0.81 \[ - \sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )} + x \sqrt{a + \frac{b}{x^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b/x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0504626, size = 71, normalized size = 1.69 \[ \frac{x \sqrt{a+\frac{b}{x^2}} \left (\sqrt{a x^2+b}-\sqrt{b} \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )+\sqrt{b} \log (x)\right )}{\sqrt{a x^2+b}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x^2],x]
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Maple [A] time = 0.008, size = 63, normalized size = 1.5 \[ -{x\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( \sqrt{b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ) -\sqrt{a{x}^{2}+b} \right ){\frac{1}{\sqrt{a{x}^{2}+b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b/x^2)^(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(sqrt(a + b/x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.251483, size = 1, normalized size = 0.02 \[ \left [x \sqrt{\frac{a x^{2} + b}{x^{2}}} + \frac{1}{2} \, \sqrt{b} \log \left (-\frac{a x^{2} - 2 \, \sqrt{b} x \sqrt{\frac{a x^{2} + b}{x^{2}}} + 2 \, b}{x^{2}}\right ), x \sqrt{\frac{a x^{2} + b}{x^{2}}} - \sqrt{-b} \arctan \left (\frac{b}{\sqrt{-b} x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.27006, size = 56, normalized size = 1.33 \[ \frac{\sqrt{a} x}{\sqrt{1 + \frac{b}{a x^{2}}}} - \sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )} + \frac{b}{\sqrt{a} x \sqrt{1 + \frac{b}{a x^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a+b/x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.233782, size = 92, normalized size = 2.19 \[{\left (\frac{b \arctan \left (\frac{\sqrt{a x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} + \sqrt{a x^{2} + b}\right )}{\rm sign}\left (x\right ) - \frac{{\left (b \arctan \left (\frac{\sqrt{b}}{\sqrt{-b}}\right ) + \sqrt{-b} \sqrt{b}\right )}{\rm sign}\left (x\right )}{\sqrt{-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^2),x, algorithm="giac")
[Out]