Optimal. Leaf size=50 \[ \frac{1}{4} x^2 \sqrt{a+c x^4}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{4 \sqrt{c}} \]
[Out]
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Rubi [A] time = 0.0561052, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{1}{4} x^2 \sqrt{a+c x^4}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{4 \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[a + c*x^4],x]
[Out]
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Rubi in Sympy [A] time = 5.11774, size = 42, normalized size = 0.84 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{4 \sqrt{c}} + \frac{x^{2} \sqrt{a + c x^{4}}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(c*x**4+a)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0293181, size = 53, normalized size = 1.06 \[ \frac{1}{4} x^2 \sqrt{a+c x^4}+\frac{a \log \left (\sqrt{c} \sqrt{a+c x^4}+c x^2\right )}{4 \sqrt{c}} \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[a + c*x^4],x]
[Out]
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Maple [A] time = 0.009, size = 40, normalized size = 0.8 \[{\frac{{x}^{2}}{4}\sqrt{c{x}^{4}+a}}+{\frac{a}{4}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){\frac{1}{\sqrt{c}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(c*x^4+a)^(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(sqrt(c*x^4 + a)*x,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.29485, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{c x^{4} + a} \sqrt{c} x^{2} + a \log \left (-2 \, \sqrt{c x^{4} + a} c x^{2} -{\left (2 \, c x^{4} + a\right )} \sqrt{c}\right )}{8 \, \sqrt{c}}, \frac{\sqrt{c x^{4} + a} \sqrt{-c} x^{2} + a \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right )}{4 \, \sqrt{-c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + a)*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 6.24094, size = 44, normalized size = 0.88 \[ \frac{\sqrt{a} x^{2} \sqrt{1 + \frac{c x^{4}}{a}}}{4} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(c*x**4+a)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.217435, size = 55, normalized size = 1.1 \[ \frac{1}{4} \, \sqrt{c x^{4} + a} x^{2} - \frac{a{\rm ln}\left ({\left | -\sqrt{c} x^{2} + \sqrt{c x^{4} + a} \right |}\right )}{4 \, \sqrt{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + a)*x,x, algorithm="giac")
[Out]