Optimal. Leaf size=49 \[ \frac{1}{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{\sqrt{a+c x^4}}{2 x^2} \]
[Out]
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Rubi [A] time = 0.0665315, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{1}{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{\sqrt{a+c x^4}}{2 x^2} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[a + c*x^4]/x^3,x]
[Out]
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Rubi in Sympy [A] time = 7.07481, size = 41, normalized size = 0.84 \[ \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{2} - \frac{\sqrt{a + c x^{4}}}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+a)**(1/2)/x**3,x)
[Out]
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Mathematica [A] time = 0.0607901, size = 49, normalized size = 1. \[ \frac{1}{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{\sqrt{a+c x^4}}{2 x^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[a + c*x^4]/x^3,x]
[Out]
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Maple [A] time = 0.015, size = 60, normalized size = 1.2 \[ -{\frac{1}{2\,a{x}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{c{x}^{2}}{2\,a}\sqrt{c{x}^{4}+a}}+{\frac{1}{2}\sqrt{c}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+a)^(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 + a)/x^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.267745, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{c} x^{2} \log \left (-2 \, c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{c} x^{2} - a\right ) - 2 \, \sqrt{c x^{4} + a}}{4 \, x^{2}}, \frac{\sqrt{-c} x^{2} \arctan \left (\frac{c x^{2}}{\sqrt{c x^{4} + a} \sqrt{-c}}\right ) - \sqrt{c x^{4} + a}}{2 \, x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + a)/x^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.94525, size = 66, normalized size = 1.35 \[ - \frac{\sqrt{a}}{2 x^{2} \sqrt{1 + \frac{c x^{4}}{a}}} + \frac{\sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2} - \frac{c x^{2}}{2 \sqrt{a} \sqrt{1 + \frac{c x^{4}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+a)**(1/2)/x**3,x)
[Out]
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GIAC/XCAS [A] time = 0.219232, size = 49, normalized size = 1. \[ -\frac{c \arctan \left (\frac{\sqrt{c + \frac{a}{x^{4}}}}{\sqrt{-c}}\right )}{2 \, \sqrt{-c}} - \frac{1}{2} \, \sqrt{c + \frac{a}{x^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + a)/x^3,x, algorithm="giac")
[Out]