Optimal. Leaf size=47 \[ \frac{1}{4} x^4 \sqrt{a+\frac{b}{x^4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]
[Out]
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Rubi [A] time = 0.0869045, antiderivative size = 47, 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 \[ \frac{1}{4} x^4 \sqrt{a+\frac{b}{x^4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + b/x^4]*x^3,x]
[Out]
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Rubi in Sympy [A] time = 6.9488, size = 39, normalized size = 0.83 \[ \frac{x^{4} \sqrt{a + \frac{b}{x^{4}}}}{4} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{4 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(a+b/x**4)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0676393, size = 64, normalized size = 1.36 \[ \frac{1}{4} x^2 \sqrt{a+\frac{b}{x^4}} \left (\frac{b \log \left (\sqrt{a} \sqrt{a x^4+b}+a x^2\right )}{\sqrt{a} \sqrt{a x^4+b}}+x^2\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + b/x^4]*x^3,x]
[Out]
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Maple [A] time = 0.024, size = 68, normalized size = 1.5 \[{\frac{{x}^{2}}{4}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ({x}^{2}\sqrt{a{x}^{4}+b}\sqrt{a}+b\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ) \right ){\frac{1}{\sqrt{a{x}^{4}+b}}}{\frac{1}{\sqrt{a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(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^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254519, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} + \sqrt{a} b \log \left (-2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (2 \, a x^{4} + b\right )} \sqrt{a}\right )}{8 \, a}, \frac{a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - \sqrt{-a} b \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}\right )}{4 \, a}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^4)*x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.70102, size = 44, normalized size = 0.94 \[ \frac{\sqrt{b} x^{2} \sqrt{\frac{a x^{4}}{b} + 1}}{4} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{4 \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(a+b/x**4)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.232942, size = 55, normalized size = 1.17 \[ \frac{1}{4} \, \sqrt{a x^{4} + b} x^{2} - \frac{b{\rm ln}\left ({\left | -\sqrt{a} x^{2} + \sqrt{a x^{4} + b} \right |}\right )}{4 \, \sqrt{a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(a + b/x^4)*x^3,x, algorithm="giac")
[Out]