Optimal. Leaf size=77 \[ \frac{\sqrt{-a+b x^2+c x^4}}{2 a x^2}-\frac{b \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{4 a^{3/2}} \]
[Out]
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Rubi [A] time = 0.141881, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\sqrt{-a+b x^2+c x^4}}{2 a x^2}-\frac{b \tan ^{-1}\left (\frac{2 a-b x^2}{2 \sqrt{a} \sqrt{-a+b x^2+c x^4}}\right )}{4 a^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*Sqrt[-a + b*x^2 + c*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 15.1498, size = 61, normalized size = 0.79 \[ \frac{\sqrt{- a + b x^{2} + c x^{4}}}{2 a x^{2}} + \frac{b \operatorname{atan}{\left (\frac{- 2 a + b x^{2}}{2 \sqrt{a} \sqrt{- a + b x^{2} + c x^{4}}} \right )}}{4 a^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(c*x**4+b*x**2-a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.118517, size = 93, normalized size = 1.21 \[ -\frac{2 \sqrt{-a} \sqrt{-a+b x^2+c x^4}-b x^2 \log \left (2 \sqrt{-a} \sqrt{-a+b x^2+c x^4}-2 a+b x^2\right )+2 b x^2 \log (x)}{4 (-a)^{3/2} x^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*Sqrt[-a + b*x^2 + c*x^4]),x]
[Out]
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Maple [A] time = 0.019, size = 74, normalized size = 1. \[{\frac{1}{2\,a{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}-a}}-{\frac{b}{4\,a}\ln \left ({\frac{1}{{x}^{2}} \left ( -2\,a+b{x}^{2}+2\,\sqrt{-a}\sqrt{c{x}^{4}+b{x}^{2}-a} \right ) } \right ){\frac{1}{\sqrt{-a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(c*x^4+b*x^2-a)^(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(1/(sqrt(c*x^4 + b*x^2 - a)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.293889, size = 1, normalized size = 0.01 \[ \left [\frac{b x^{2} \log \left (\frac{4 \, \sqrt{c x^{4} + b x^{2} - a}{\left (a b x^{2} - 2 \, a^{2}\right )} +{\left ({\left (b^{2} - 4 \, a c\right )} x^{4} - 8 \, a b x^{2} + 8 \, a^{2}\right )} \sqrt{-a}}{x^{4}}\right ) + 4 \, \sqrt{c x^{4} + b x^{2} - a} \sqrt{-a}}{8 \, \sqrt{-a} a x^{2}}, \frac{b x^{2} \arctan \left (\frac{b x^{2} - 2 \, a}{2 \, \sqrt{c x^{4} + b x^{2} - a} \sqrt{a}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2} - a} \sqrt{a}}{4 \, a^{\frac{3}{2}} x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 + b*x^2 - a)*x^3),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{3} \sqrt{- a + b x^{2} + c x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(c*x**4+b*x**2-a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.324625, size = 93, normalized size = 1.21 \[ \frac{b{\rm ln}\left ({\left | -2 \, \sqrt{-a}{\left (\sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}} - \frac{\sqrt{-a}}{x^{2}}\right )} + b \right |}\right )}{4 \, \sqrt{-a} a} + \frac{\sqrt{c + \frac{b}{x^{2}} - \frac{a}{x^{4}}}}{2 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 + b*x^2 - a)*x^3),x, algorithm="giac")
[Out]