Optimal. Leaf size=59 \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac{\sqrt{b x^2+c x^4}}{2 b x^3} \]
[Out]
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Rubi [A] time = 0.0965936, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 b^{3/2}}-\frac{\sqrt{b x^2+c x^4}}{2 b x^3} \]
Antiderivative was successfully verified.
[In] Int[1/(x^2*Sqrt[b*x^2 + c*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 13.587, size = 49, normalized size = 0.83 \[ - \frac{\sqrt{b x^{2} + c x^{4}}}{2 b x^{3}} + \frac{c \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{2 b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**2/(c*x**4+b*x**2)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0728432, size = 97, normalized size = 1.64 \[ \frac{-\sqrt{b} \left (b+c x^2\right )-c x^2 \log (x) \sqrt{b+c x^2}+c x^2 \sqrt{b+c x^2} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )}{2 b^{3/2} x \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^2*Sqrt[b*x^2 + c*x^4]),x]
[Out]
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Maple [A] time = 0.011, size = 73, normalized size = 1.2 \[{\frac{1}{2\,x}\sqrt{c{x}^{2}+b} \left ( c\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){x}^{2}b-\sqrt{c{x}^{2}+b}{b}^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{b}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^2/(c*x^4+b*x^2)^(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)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.276566, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b} c x^{3} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} + 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}} b}{4 \, b^{2} x^{3}}, -\frac{\sqrt{-b} c x^{3} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}} b}{2 \, b^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 + b*x^2)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{x^{2} \left (b + c x^{2}\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**2/(c*x**4+b*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^4 + b*x^2)*x^2),x, algorithm="giac")
[Out]