Optimal. Leaf size=114 \[ \frac{\sqrt{x^8+1} x^2}{2 \left (x^4+1\right )}+\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{4 \sqrt{x^8+1}}-\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{2 \sqrt{x^8+1}} \]
[Out]
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Rubi [A] time = 0.10884, antiderivative size = 114, 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 \[ \frac{\sqrt{x^8+1} x^2}{2 \left (x^4+1\right )}+\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{4 \sqrt{x^8+1}}-\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{2 \sqrt{x^8+1}} \]
Antiderivative was successfully verified.
[In] Int[x^5/Sqrt[1 + x^8],x]
[Out]
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Rubi in Sympy [A] time = 7.63435, size = 99, normalized size = 0.87 \[ \frac{x^{2} \sqrt{x^{8} + 1}}{2 \left (x^{4} + 1\right )} - \frac{\sqrt{\frac{x^{8} + 1}{\left (x^{4} + 1\right )^{2}}} \left (x^{4} + 1\right ) E\left (2 \operatorname{atan}{\left (x^{2} \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{x^{8} + 1}} + \frac{\sqrt{\frac{x^{8} + 1}{\left (x^{4} + 1\right )^{2}}} \left (x^{4} + 1\right ) F\left (2 \operatorname{atan}{\left (x^{2} \right )}\middle | \frac{1}{2}\right )}{4 \sqrt{x^{8} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(x**8+1)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0217739, size = 22, normalized size = 0.19 \[ \frac{1}{6} x^6 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-x^8\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^5/Sqrt[1 + x^8],x]
[Out]
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Maple [C] time = 0.023, size = 17, normalized size = 0.2 \[{\frac{{x}^{6}}{6}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{7}{4}};\,-{x}^{8})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(x^8+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{x^{8} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(x^8 + 1),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{5}}{\sqrt{x^{8} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(x^8 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.06233, size = 29, normalized size = 0.25 \[ \frac{x^{6} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(x**8+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{5}}{\sqrt{x^{8} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/sqrt(x^8 + 1),x, algorithm="giac")
[Out]