3.1524 \(\int \frac{x^5}{\sqrt{1+x^8}} \, dx\)

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]

(x^2*Sqrt[1 + x^8])/(2*(1 + x^4)) - ((1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4)^2]*Ellip
ticE[2*ArcTan[x^2], 1/2])/(2*Sqrt[1 + x^8]) + ((1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4
)^2]*EllipticF[2*ArcTan[x^2], 1/2])/(4*Sqrt[1 + x^8])

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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]

(x^2*Sqrt[1 + x^8])/(2*(1 + x^4)) - ((1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4)^2]*Ellip
ticE[2*ArcTan[x^2], 1/2])/(2*Sqrt[1 + x^8]) + ((1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4
)^2]*EllipticF[2*ArcTan[x^2], 1/2])/(4*Sqrt[1 + x^8])

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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]

x**2*sqrt(x**8 + 1)/(2*(x**4 + 1)) - sqrt((x**8 + 1)/(x**4 + 1)**2)*(x**4 + 1)*e
lliptic_e(2*atan(x**2), 1/2)/(2*sqrt(x**8 + 1)) + sqrt((x**8 + 1)/(x**4 + 1)**2)
*(x**4 + 1)*elliptic_f(2*atan(x**2), 1/2)/(4*sqrt(x**8 + 1))

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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]

(x^6*Hypergeometric2F1[1/2, 3/4, 7/4, -x^8])/6

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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]

1/6*x^6*hypergeom([1/2,3/4],[7/4],-x^8)

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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]

integrate(x^5/sqrt(x^8 + 1), x)

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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]

integral(x^5/sqrt(x^8 + 1), x)

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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]

x**6*gamma(3/4)*hyper((1/2, 3/4), (7/4,), x**8*exp_polar(I*pi))/(8*gamma(7/4))

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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]

integrate(x^5/sqrt(x^8 + 1), x)