Optimal. Leaf size=42 \[ x \sqrt{\frac{a}{x^2}+b}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b}}\right ) \]
[Out]
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Rubi [A] time = 0.0618879, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ x \sqrt{\frac{a}{x^2}+b}-\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^2}+b}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[(a + b*x^2)/x^2],x]
[Out]
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Rubi in Sympy [A] time = 4.93287, size = 34, normalized size = 0.81 \[ - \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a}}{x \sqrt{\frac{a}{x^{2}} + b}} \right )} + x \sqrt{\frac{a}{x^{2}} + b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(((b*x**2+a)/x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0469191, size = 71, normalized size = 1.69 \[ \frac{x \sqrt{\frac{a}{x^2}+b} \left (\sqrt{a+b x^2}-\sqrt{a} \log \left (\sqrt{a} \sqrt{a+b x^2}+a\right )+\sqrt{a} \log (x)\right )}{\sqrt{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[(a + b*x^2)/x^2],x]
[Out]
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Maple [A] time = 0.01, size = 61, normalized size = 1.5 \[{x\sqrt{{\frac{b{x}^{2}+a}{{x}^{2}}}} \left ( \sqrt{b{x}^{2}+a}-\sqrt{a}\ln \left ( 2\,{\frac{\sqrt{a}\sqrt{b{x}^{2}+a}+a}{x}} \right ) \right ){\frac{1}{\sqrt{b{x}^{2}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(((b*x^2+a)/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((b*x^2 + a)/x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.251397, size = 1, normalized size = 0.02 \[ \left [x \sqrt{\frac{b x^{2} + a}{x^{2}}} + \frac{1}{2} \, \sqrt{a} \log \left (-\frac{b x^{2} - 2 \, \sqrt{a} x \sqrt{\frac{b x^{2} + a}{x^{2}}} + 2 \, a}{x^{2}}\right ), x \sqrt{\frac{b x^{2} + a}{x^{2}}} - \sqrt{-a} \arctan \left (\frac{a}{\sqrt{-a} x \sqrt{\frac{b x^{2} + a}{x^{2}}}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)/x^2),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x**2+a)/x**2)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.222623, size = 92, normalized size = 2.19 \[{\left (\frac{a \arctan \left (\frac{\sqrt{b x^{2} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + \sqrt{b x^{2} + a}\right )}{\rm sign}\left (x\right ) - \frac{{\left (a \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) + \sqrt{-a} \sqrt{a}\right )}{\rm sign}\left (x\right )}{\sqrt{-a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt((b*x^2 + a)/x^2),x, algorithm="giac")
[Out]