Optimal. Leaf size=51 \[ \frac{x}{b \sqrt{b x^2+c x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{b^{3/2}} \]
[Out]
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Rubi [A] time = 0.1007, antiderivative size = 51, 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 \[ \frac{x}{b \sqrt{b x^2+c x^4}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{b^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^2/(b*x^2 + c*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 13.8143, size = 42, normalized size = 0.82 \[ \frac{x}{b \sqrt{b x^{2} + c x^{4}}} - \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(c*x**4+b*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0590084, size = 75, normalized size = 1.47 \[ \frac{x \left (\log (x) \sqrt{b+c x^2}-\sqrt{b+c x^2} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )+\sqrt{b}\right )}{b^{3/2} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(b*x^2 + c*x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.008, size = 67, normalized size = 1.3 \[ -{{x}^{3} \left ( c{x}^{2}+b \right ) \left ( \ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) b\sqrt{c{x}^{2}+b}-{b}^{{\frac{3}{2}}} \right ) \left ( c{x}^{4}+b{x}^{2} \right ) ^{-{\frac{3}{2}}}{b}^{-{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(c*x^4+b*x^2)^(3/2),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(x^2/(c*x^4 + b*x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.275976, size = 1, normalized size = 0.02 \[ \left [\frac{{\left (c x^{3} + b x\right )} \sqrt{b} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) + 2 \, \sqrt{c x^{4} + b x^{2}} b}{2 \,{\left (b^{2} c x^{3} + b^{3} x\right )}}, \frac{{\left (c x^{3} + b x\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) + \sqrt{c x^{4} + b x^{2}} b}{b^{2} c x^{3} + b^{3} x}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(c*x^4 + b*x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(c*x**4+b*x**2)**(3/2),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(x^2/(c*x^4 + b*x^2)^(3/2),x, algorithm="giac")
[Out]