3.1522 \(\int \frac{x^{13}}{\sqrt{1+x^8}} \, dx\)

Optimal. Leaf size=130 \[ \frac{1}{10} \sqrt{x^8+1} x^6-\frac{3 \sqrt{x^8+1} x^2}{10 \left (x^4+1\right )}-\frac{3 \left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{20 \sqrt{x^8+1}}+\frac{3 \left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{10 \sqrt{x^8+1}} \]

[Out]

(x^6*Sqrt[1 + x^8])/10 - (3*x^2*Sqrt[1 + x^8])/(10*(1 + x^4)) + (3*(1 + x^4)*Sqr
t[(1 + x^8)/(1 + x^4)^2]*EllipticE[2*ArcTan[x^2], 1/2])/(10*Sqrt[1 + x^8]) - (3*
(1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4)^2]*EllipticF[2*ArcTan[x^2], 1/2])/(20*Sqrt[1
+ x^8])

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Rubi [A]  time = 0.137168, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385 \[ \frac{1}{10} \sqrt{x^8+1} x^6-\frac{3 \sqrt{x^8+1} x^2}{10 \left (x^4+1\right )}-\frac{3 \left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{20 \sqrt{x^8+1}}+\frac{3 \left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{10 \sqrt{x^8+1}} \]

Antiderivative was successfully verified.

[In]  Int[x^13/Sqrt[1 + x^8],x]

[Out]

(x^6*Sqrt[1 + x^8])/10 - (3*x^2*Sqrt[1 + x^8])/(10*(1 + x^4)) + (3*(1 + x^4)*Sqr
t[(1 + x^8)/(1 + x^4)^2]*EllipticE[2*ArcTan[x^2], 1/2])/(10*Sqrt[1 + x^8]) - (3*
(1 + x^4)*Sqrt[(1 + x^8)/(1 + x^4)^2]*EllipticF[2*ArcTan[x^2], 1/2])/(20*Sqrt[1
+ x^8])

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Rubi in Sympy [A]  time = 9.35055, size = 117, normalized size = 0.9 \[ \frac{x^{6} \sqrt{x^{8} + 1}}{10} - \frac{3 x^{2} \sqrt{x^{8} + 1}}{10 \left (x^{4} + 1\right )} + \frac{3 \sqrt{\frac{x^{8} + 1}{\left (x^{4} + 1\right )^{2}}} \left (x^{4} + 1\right ) E\left (2 \operatorname{atan}{\left (x^{2} \right )}\middle | \frac{1}{2}\right )}{10 \sqrt{x^{8} + 1}} - \frac{3 \sqrt{\frac{x^{8} + 1}{\left (x^{4} + 1\right )^{2}}} \left (x^{4} + 1\right ) F\left (2 \operatorname{atan}{\left (x^{2} \right )}\middle | \frac{1}{2}\right )}{20 \sqrt{x^{8} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**13/(x**8+1)**(1/2),x)

[Out]

x**6*sqrt(x**8 + 1)/10 - 3*x**2*sqrt(x**8 + 1)/(10*(x**4 + 1)) + 3*sqrt((x**8 +
1)/(x**4 + 1)**2)*(x**4 + 1)*elliptic_e(2*atan(x**2), 1/2)/(10*sqrt(x**8 + 1)) -
3*sqrt((x**8 + 1)/(x**4 + 1)**2)*(x**4 + 1)*elliptic_f(2*atan(x**2), 1/2)/(20*s
qrt(x**8 + 1))

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Mathematica [C]  time = 0.0292938, size = 34, normalized size = 0.26 \[ \frac{1}{10} x^6 \left (\sqrt{x^8+1}-\, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-x^8\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^13/Sqrt[1 + x^8],x]

[Out]

(x^6*(Sqrt[1 + x^8] - Hypergeometric2F1[1/2, 3/4, 7/4, -x^8]))/10

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Maple [C]  time = 0.036, size = 30, normalized size = 0.2 \[{\frac{{x}^{6}}{10}\sqrt{{x}^{8}+1}}-{\frac{{x}^{6}}{10}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{7}{4}};\,-{x}^{8})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^13/(x^8+1)^(1/2),x)

[Out]

1/10*x^6*(x^8+1)^(1/2)-1/10*x^6*hypergeom([1/2,3/4],[7/4],-x^8)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{\sqrt{x^{8} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/sqrt(x^8 + 1),x, algorithm="maxima")

[Out]

integrate(x^13/sqrt(x^8 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{13}}{\sqrt{x^{8} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/sqrt(x^8 + 1),x, algorithm="fricas")

[Out]

integral(x^13/sqrt(x^8 + 1), x)

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Sympy [A]  time = 6.22477, size = 29, normalized size = 0.22 \[ \frac{x^{14} \Gamma \left (\frac{7}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{11}{4} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 \Gamma \left (\frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**13/(x**8+1)**(1/2),x)

[Out]

x**14*gamma(7/4)*hyper((1/2, 7/4), (11/4,), x**8*exp_polar(I*pi))/(8*gamma(11/4)
)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{13}}{\sqrt{x^{8} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^13/sqrt(x^8 + 1),x, algorithm="giac")

[Out]

integrate(x^13/sqrt(x^8 + 1), x)