Optimal. Leaf size=53 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{3/2}}-\frac{\sqrt{a+\frac{b}{x^2}}}{2 b x} \]
[Out]
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Rubi [A] time = 0.0777898, antiderivative size = 53, 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 \[ \frac{a \tanh ^{-1}\left (\frac{\sqrt{b}}{x \sqrt{a+\frac{b}{x^2}}}\right )}{2 b^{3/2}}-\frac{\sqrt{a+\frac{b}{x^2}}}{2 b x} \]
Antiderivative was successfully verified.
[In] Int[1/(Sqrt[a + b/x^2]*x^4),x]
[Out]
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Rubi in Sympy [A] time = 7.40605, size = 41, normalized size = 0.77 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{b}}{x \sqrt{a + \frac{b}{x^{2}}}} \right )}}{2 b^{\frac{3}{2}}} - \frac{\sqrt{a + \frac{b}{x^{2}}}}{2 b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**2)**(1/2)/x**4,x)
[Out]
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Mathematica [A] time = 0.0749784, size = 93, normalized size = 1.75 \[ \frac{-\sqrt{b} \left (a x^2+b\right )-a x^2 \log (x) \sqrt{a x^2+b}+a x^2 \sqrt{a x^2+b} \log \left (\sqrt{b} \sqrt{a x^2+b}+b\right )}{2 b^{3/2} x^3 \sqrt{a+\frac{b}{x^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[a + b/x^2]*x^4),x]
[Out]
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Maple [A] time = 0.01, size = 73, normalized size = 1.4 \[ -{\frac{1}{2\,{x}^{3}}\sqrt{a{x}^{2}+b} \left ( -a\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{a{x}^{2}+b}+b}{x}} \right ){x}^{2}b+\sqrt{a{x}^{2}+b}{b}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}}}}{b}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^2)^(1/2)/x^4,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/(sqrt(a + b/x^2)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.251512, size = 1, normalized size = 0.02 \[ \left [\frac{a \sqrt{b} x \log \left (-\frac{2 \, b x \sqrt{\frac{a x^{2} + b}{x^{2}}} +{\left (a x^{2} + 2 \, b\right )} \sqrt{b}}{x^{2}}\right ) - 2 \, b \sqrt{\frac{a x^{2} + b}{x^{2}}}}{4 \, b^{2} x}, -\frac{a \sqrt{-b} x \arctan \left (\frac{\sqrt{-b}}{x \sqrt{\frac{a x^{2} + b}{x^{2}}}}\right ) + b \sqrt{\frac{a x^{2} + b}{x^{2}}}}{2 \, b^{2} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^2)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.00321, size = 42, normalized size = 0.79 \[ - \frac{\sqrt{a} \sqrt{1 + \frac{b}{a x^{2}}}}{2 b x} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b}}{\sqrt{a} x} \right )}}{2 b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**2)**(1/2)/x**4,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{2}}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(a + b/x^2)*x^4),x, algorithm="giac")
[Out]