Optimal. Leaf size=62 \[ \frac{1}{6} x^2 \sqrt{x^8+1}-\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{12 \sqrt{x^8+1}} \]
[Out]
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Rubi [A] time = 0.0692216, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1}{6} x^2 \sqrt{x^8+1}-\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{12 \sqrt{x^8+1}} \]
Antiderivative was successfully verified.
[In] Int[x^9/Sqrt[1 + x^8],x]
[Out]
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Rubi in Sympy [A] time = 4.67807, size = 53, normalized size = 0.85 \[ \frac{x^{2} \sqrt{x^{8} + 1}}{6} - \frac{\sqrt{\frac{x^{8} + 1}{\left (x^{4} + 1\right )^{2}}} \left (x^{4} + 1\right ) F\left (2 \operatorname{atan}{\left (x^{2} \right )}\middle | \frac{1}{2}\right )}{12 \sqrt{x^{8} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**9/(x**8+1)**(1/2),x)
[Out]
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Mathematica [C] time = 0.0284059, size = 34, normalized size = 0.55 \[ \frac{1}{6} x^2 \left (\sqrt{x^8+1}-\, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-x^8\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^9/Sqrt[1 + x^8],x]
[Out]
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Maple [C] time = 0.033, size = 30, normalized size = 0.5 \[{\frac{{x}^{2}}{6}\sqrt{{x}^{8}+1}}-{\frac{{x}^{2}}{6}{\mbox{$_2$F$_1$}({\frac{1}{4}},{\frac{1}{2}};\,{\frac{5}{4}};\,-{x}^{8})}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^9/(x^8+1)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{\sqrt{x^{8} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^9/sqrt(x^8 + 1),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{9}}{\sqrt{x^{8} + 1}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^9/sqrt(x^8 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.17253, size = 29, normalized size = 0.47 \[ \frac{x^{10} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**9/(x**8+1)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{9}}{\sqrt{x^{8} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^9/sqrt(x^8 + 1),x, algorithm="giac")
[Out]