3.1510 \(\int \frac{\sqrt{1+x^8}}{x^3} \, dx\)

Optimal. Leaf size=125 \[ -\frac{\sqrt{x^8+1}}{2 x^2}+\frac{\sqrt{x^8+1} x^2}{x^4+1}+\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{2 \sqrt{x^8+1}}-\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{\sqrt{x^8+1}} \]

[Out]

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

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Rubi [A]  time = 0.134705, antiderivative size = 125, 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 \[ -\frac{\sqrt{x^8+1}}{2 x^2}+\frac{\sqrt{x^8+1} x^2}{x^4+1}+\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} F\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{2 \sqrt{x^8+1}}-\frac{\left (x^4+1\right ) \sqrt{\frac{x^8+1}{\left (x^4+1\right )^2}} E\left (2 \tan ^{-1}\left (x^2\right )|\frac{1}{2}\right )}{\sqrt{x^8+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x^8]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.31045, size = 109, normalized size = 0.87 \[ \frac{x^{2} \sqrt{x^{8} + 1}}{x^{4} + 1} - \frac{\sqrt{\frac{x^{8} + 1}{\left (x^{4} + 1\right )^{2}}} \left (x^{4} + 1\right ) E\left (2 \operatorname{atan}{\left (x^{2} \right )}\middle | \frac{1}{2}\right )}{\sqrt{x^{8} + 1}} + \frac{\sqrt{\frac{x^{8} + 1}{\left (x^{4} + 1\right )^{2}}} \left (x^{4} + 1\right ) F\left (2 \operatorname{atan}{\left (x^{2} \right )}\middle | \frac{1}{2}\right )}{2 \sqrt{x^{8} + 1}} - \frac{\sqrt{x^{8} + 1}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**8+1)**(1/2)/x**3,x)

[Out]

x**2*sqrt(x**8 + 1)/(x**4 + 1) - sqrt((x**8 + 1)/(x**4 + 1)**2)*(x**4 + 1)*ellip
tic_e(2*atan(x**2), 1/2)/sqrt(x**8 + 1) + sqrt((x**8 + 1)/(x**4 + 1)**2)*(x**4 +
1)*elliptic_f(2*atan(x**2), 1/2)/(2*sqrt(x**8 + 1)) - sqrt(x**8 + 1)/(2*x**2)

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Mathematica [C]  time = 0.0288865, size = 39, normalized size = 0.31 \[ \frac{1}{3} x^6 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-x^8\right )-\frac{\sqrt{x^8+1}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + x^8]/x^3,x]

[Out]

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

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Maple [C]  time = 0.047, size = 30, normalized size = 0.2 \[ -{\frac{1}{2\,{x}^{2}}\sqrt{{x}^{8}+1}}+{\frac{{x}^{6}}{3}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{3}{4}};\,{\frac{7}{4}};\,-{x}^{8})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^8+1)^(1/2)/x^3,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{8} + 1}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^8 + 1)/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(x^8 + 1)/x^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{x^{8} + 1}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^8 + 1)/x^3,x, algorithm="fricas")

[Out]

integral(sqrt(x^8 + 1)/x^3, x)

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Sympy [A]  time = 1.9858, size = 34, normalized size = 0.27 \[ \frac{\Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{x^{8} e^{i \pi }} \right )}}{8 x^{2} \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**8+1)**(1/2)/x**3,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{8} + 1}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^8 + 1)/x^3,x, algorithm="giac")

[Out]

integrate(sqrt(x^8 + 1)/x^3, x)