Optimal. Leaf size=49 \[ \frac{1}{2} x^2 \sqrt{a+\frac{b}{x^4}}-\frac{1}{2} \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right ) \]
[Out]
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Rubi [A] time = 0.106822, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{1}{2} x^2 \sqrt{a+\frac{b}{x^4}}-\frac{1}{2} \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b}}{x^2 \sqrt{a+\frac{b}{x^4}}}\right ) \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x^4]*x,x]
[Out]
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Rubi in Sympy [A] time = 8.44782, size = 41, normalized size = 0.84 \[ - \frac{\sqrt{b} \operatorname{atanh}{\left (\frac{\sqrt{b}}{x^{2} \sqrt{a + \frac{b}{x^{4}}}} \right )}}{2} + \frac{x^{2} \sqrt{a + \frac{b}{x^{4}}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(a+b/x**4)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0908435, size = 66, normalized size = 1.35 \[ \frac{x^2 \sqrt{a+\frac{b}{x^4}} \left (\sqrt{a x^4+b}-\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{a x^4+b}}{\sqrt{b}}\right )\right )}{2 \sqrt{a x^4+b}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x^4]*x,x]
[Out]
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Maple [A] time = 0.017, size = 65, normalized size = 1.3 \[ -{\frac{{x}^{2}}{2}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ( \sqrt{b}\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{4}+b}+b}{{x}^{2}}} \right ) -\sqrt{a{x}^{4}+b} \right ){\frac{1}{\sqrt{a{x}^{4}+b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(a+b/x^4)^(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^4)*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.247167, size = 1, normalized size = 0.02 \[ \left [\frac{1}{2} \, x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} + \frac{1}{4} \, \sqrt{b} \log \left (\frac{a x^{4} - 2 \, \sqrt{b} x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} + 2 \, b}{x^{4}}\right ), \frac{1}{2} \, x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}} - \frac{1}{2} \, \sqrt{-b} \arctan \left (\frac{x^{2} \sqrt{\frac{a x^{4} + b}{x^{4}}}}{\sqrt{-b}}\right )\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^4)*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.19021, size = 66, normalized size = 1.35 \[ \frac{\sqrt{a} x^{2}}{2 \sqrt{1 + \frac{b}{a x^{4}}}} - \frac{\sqrt{b} \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x^{2}} \right )}}{2} + \frac{b}{2 \sqrt{a} x^{2} \sqrt{1 + \frac{b}{a x^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(a+b/x**4)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.226718, size = 49, normalized size = 1. \[ \frac{b \arctan \left (\frac{\sqrt{a x^{4} + b}}{\sqrt{-b}}\right )}{2 \, \sqrt{-b}} + \frac{1}{2} \, \sqrt{a x^{4} + b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^4)*x,x, algorithm="giac")
[Out]