Optimal. Leaf size=46 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{1}{2 a \sqrt{a+\frac{b}{x^4}}} \]
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Rubi [A] time = 0.0879748, antiderivative size = 46, 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 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{2 a^{3/2}}-\frac{1}{2 a \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^4)^(3/2)*x),x]
[Out]
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Rubi in Sympy [A] time = 7.12803, size = 37, normalized size = 0.8 \[ - \frac{1}{2 a \sqrt{a + \frac{b}{x^{4}}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{2 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**4)**(3/2)/x,x)
[Out]
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Mathematica [A] time = 0.0398852, size = 70, normalized size = 1.52 \[ \frac{\sqrt{a x^4+b} \log \left (\sqrt{a} \sqrt{a x^4+b}+a x^2\right )-\sqrt{a} x^2}{2 a^{3/2} x^2 \sqrt{a+\frac{b}{x^4}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^4)^(3/2)*x),x]
[Out]
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Maple [A] time = 0.02, size = 67, normalized size = 1.5 \[{\frac{a{x}^{4}+b}{2\,{x}^{6}} \left ( -{x}^{2}{a}^{{\frac{3}{2}}}+\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ) a\sqrt{a{x}^{4}+b} \right ) \left ({\frac{a{x}^{4}+b}{{x}^{4}}} \right ) ^{-{\frac{3}{2}}}{a}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^4)^(3/2)/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(1/((a + b/x^4)^(3/2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.260137, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (a x^{4} + b\right )} \sqrt{a} \log \left (-2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (2 \, a x^{4} + b\right )} \sqrt{a}\right )}{4 \,{\left (a^{3} x^{4} + a^{2} b\right )}}, -\frac{a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} +{\left (a x^{4} + b\right )} \sqrt{-a} \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}\right )}{2 \,{\left (a^{3} x^{4} + a^{2} b\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^4)^(3/2)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 8.51045, size = 187, normalized size = 4.07 \[ - \frac{2 a^{3} x^{4} \sqrt{1 + \frac{b}{a x^{4}}}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} - \frac{a^{3} x^{4} \log{\left (\frac{b}{a x^{4}} \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} + \frac{2 a^{3} x^{4} \log{\left (\sqrt{1 + \frac{b}{a x^{4}}} + 1 \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} - \frac{a^{2} b \log{\left (\frac{b}{a x^{4}} \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} + \frac{2 a^{2} b \log{\left (\sqrt{1 + \frac{b}{a x^{4}}} + 1 \right )}}{4 a^{\frac{9}{2}} x^{4} + 4 a^{\frac{7}{2}} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**4)**(3/2)/x,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (a + \frac{b}{x^{4}}\right )}^{\frac{3}{2}} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^4)^(3/2)*x),x, algorithm="giac")
[Out]