Optimal. Leaf size=56 \[ -\frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 \sqrt{b}}-\frac{\sqrt{b x^2+c x^4}}{2 x^3} \]
[Out]
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Rubi [A] time = 0.0920732, antiderivative size = 56, 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 \[ -\frac{c \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{2 \sqrt{b}}-\frac{\sqrt{b x^2+c x^4}}{2 x^3} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[b*x^2 + c*x^4]/x^4,x]
[Out]
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Rubi in Sympy [A] time = 12.9372, size = 49, normalized size = 0.88 \[ - \frac{\sqrt{b x^{2} + c x^{4}}}{2 x^{3}} - \frac{c \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{2 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+b*x**2)**(1/2)/x**4,x)
[Out]
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Mathematica [A] time = 0.0651876, size = 89, normalized size = 1.59 \[ -\frac{\sqrt{x^2 \left (b+c x^2\right )} \left (\sqrt{b} \sqrt{b+c x^2}+c x^2 \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )-c x^2 \log (x)\right )}{2 \sqrt{b} x^3 \sqrt{b+c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[b*x^2 + c*x^4]/x^4,x]
[Out]
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Maple [A] time = 0.009, size = 85, normalized size = 1.5 \[ -{\frac{1}{2\,b{x}^{3}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( \ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ) \sqrt{b}{x}^{2}c-\sqrt{c{x}^{2}+b}{x}^{2}c+ \left ( c{x}^{2}+b \right ) ^{{\frac{3}{2}}} \right ){\frac{1}{\sqrt{c{x}^{2}+b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+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(sqrt(c*x^4 + b*x^2)/x^4,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.274876, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{b} c x^{3} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} - 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) - 2 \, \sqrt{c x^{4} + b x^{2}} b}{4 \, b x^{3}}, \frac{\sqrt{-b} c x^{3} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) - \sqrt{c x^{4} + b x^{2}} b}{2 \, b x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)/x^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (b + c x^{2}\right )}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+b*x**2)**(1/2)/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.289783, size = 61, normalized size = 1.09 \[ \frac{1}{2} \, c{\left (\frac{\arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{\sqrt{c x^{2} + b}}{c x^{2}}\right )}{\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)/x^4,x, algorithm="giac")
[Out]