Optimal. Leaf size=52 \[ \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )-\frac{\sqrt{b x^2+c x^4}}{x^2} \]
[Out]
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Rubi [A] time = 0.146377, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{b x^2+c x^4}}\right )-\frac{\sqrt{b x^2+c x^4}}{x^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*x^2 + c*x^4]/x^3,x]
[Out]
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Rubi in Sympy [A] time = 12.6633, size = 44, normalized size = 0.85 \[ \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{b x^{2} + c x^{4}}} \right )} - \frac{\sqrt{b x^{2} + c x^{4}}}{x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+b*x**2)**(1/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0490217, size = 66, normalized size = 1.27 \[ \frac{\sqrt{c} x \sqrt{b+c x^2} \log \left (\sqrt{c} \sqrt{b+c x^2}+c x\right )-b-c x^2}{\sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[b*x^2 + c*x^4]/x^3,x]
[Out]
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Maple [A] time = 0.009, size = 84, normalized size = 1.6 \[{\frac{1}{b{x}^{2}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( \sqrt{c{x}^{2}+b}{c}^{{\frac{3}{2}}}{x}^{2}+\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+b} \right ) xbc- \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}}\sqrt{c} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}{\frac{1}{\sqrt{c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+b*x^2)^(1/2)/x^3,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(sqrt(c*x^4 + b*x^2)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.276618, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{c} x^{2} \log \left (-2 \, c x^{2} - b - 2 \, \sqrt{c x^{4} + b x^{2}} \sqrt{c}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}}}{2 \, x^{2}}, \frac{\sqrt{-c} x^{2} \arctan \left (\frac{c x^{2}}{\sqrt{c x^{4} + b x^{2}} \sqrt{-c}}\right ) - \sqrt{c x^{4} + b x^{2}}}{x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)/x^3,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+b*x**2)**(1/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.293318, size = 82, normalized size = 1.58 \[ -\frac{1}{2} \, \sqrt{c}{\rm ln}\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2}\right ){\rm sign}\left (x\right ) + \frac{2 \, b \sqrt{c}{\rm sign}\left (x\right )}{{\left (\sqrt{c} x - \sqrt{c x^{2} + b}\right )}^{2} - b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)/x^3,x, algorithm="giac")
[Out]