Optimal. Leaf size=62 \[ \frac{1}{6} \sqrt{x^8+1} x^2+\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{6 \sqrt{x^8+1}} \]
[Out]
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Rubi [A] time = 0.0579022, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{1}{6} \sqrt{x^8+1} x^2+\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{6 \sqrt{x^8+1}} \]
Antiderivative was successfully verified.
[In] Int[x*Sqrt[1 + x^8],x]
[Out]
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Rubi in Sympy [A] time = 2.81271, size = 53, normalized size = 0.85 \[ \frac{x^{2} \sqrt{x^{8} + 1}}{6} + \frac{\sqrt{\frac{x^{8} + 1}{\left (x^{4} + 1\right )^{2}}} \left (x^{4} + 1\right ) F\left (2 \operatorname{atan}{\left (x^{2} \right )}\middle | \frac{1}{2}\right )}{6 \sqrt{x^{8} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*(x**8+1)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0253225, size = 34, normalized size = 0.55 \[ \frac{1}{6} x^2 \left (2 \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-x^8\right )+\sqrt{x^8+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x*Sqrt[1 + x^8],x]
[Out]
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Maple [C] time = 0.041, size = 30, normalized size = 0.5 \[{\frac{{x}^{2}}{6}\sqrt{{x}^{8}+1}}+{\frac{{x}^{2}}{3}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{5}{4}};\,-{x}^{8})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*(x^8+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{8} + 1} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^8 + 1)*x,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{x^{8} + 1} x, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^8 + 1)*x,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.74249, size = 31, normalized size = 0.5 \[ \frac{x^{2} \Gamma \left (\frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{1}{4} \\ \frac{5}{4} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{5}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*(x**8+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{8} + 1} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(x^8 + 1)*x,x, algorithm="giac")
[Out]