3.882 \(\int \frac{x^8}{\sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=43 \[ -\frac{5}{21} \sqrt{1-x^4} x-\frac{1}{7} \sqrt{1-x^4} x^5+\frac{5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

[Out]

(-5*x*Sqrt[1 - x^4])/21 - (x^5*Sqrt[1 - x^4])/7 + (5*EllipticF[ArcSin[x], -1])/2
1

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Rubi [A]  time = 0.0354781, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{5}{21} \sqrt{1-x^4} x-\frac{1}{7} \sqrt{1-x^4} x^5+\frac{5}{21} F\left (\left .\sin ^{-1}(x)\right |-1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^8/Sqrt[1 - x^4],x]

[Out]

(-5*x*Sqrt[1 - x^4])/21 - (x^5*Sqrt[1 - x^4])/7 + (5*EllipticF[ArcSin[x], -1])/2
1

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Rubi in Sympy [A]  time = 4.19139, size = 36, normalized size = 0.84 \[ - \frac{x^{5} \sqrt{- x^{4} + 1}}{7} - \frac{5 x \sqrt{- x^{4} + 1}}{21} + \frac{5 F\left (\operatorname{asin}{\left (x \right )}\middle | -1\right )}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(-x**4+1)**(1/2),x)

[Out]

-x**5*sqrt(-x**4 + 1)/7 - 5*x*sqrt(-x**4 + 1)/21 + 5*elliptic_f(asin(x), -1)/21

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Mathematica [A]  time = 0.0358736, size = 46, normalized size = 1.07 \[ \frac{3 x^9+2 x^5+5 \sqrt{1-x^4} F\left (\left .\sin ^{-1}(x)\right |-1\right )-5 x}{21 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/Sqrt[1 - x^4],x]

[Out]

(-5*x + 2*x^5 + 3*x^9 + 5*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(21*Sqrt[1 - x
^4])

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Maple [A]  time = 0.012, size = 59, normalized size = 1.4 \[ -{\frac{{x}^{5}}{7}\sqrt{-{x}^{4}+1}}-{\frac{5\,x}{21}\sqrt{-{x}^{4}+1}}+{\frac{5\,{\it EllipticF} \left ( x,i \right ) }{21}\sqrt{-{x}^{2}+1}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(-x^4+1)^(1/2),x)

[Out]

-1/7*x^5*(-x^4+1)^(1/2)-5/21*x*(-x^4+1)^(1/2)+5/21*(-x^2+1)^(1/2)*(x^2+1)^(1/2)/
(-x^4+1)^(1/2)*EllipticF(x,I)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(-x^4 + 1),x, algorithm="maxima")

[Out]

integrate(x^8/sqrt(-x^4 + 1), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{8}}{\sqrt{-x^{4} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(-x^4 + 1),x, algorithm="fricas")

[Out]

integral(x^8/sqrt(-x^4 + 1), x)

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Sympy [A]  time = 2.65064, size = 31, normalized size = 0.72 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{x^{4} e^{2 i \pi }} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(-x**4+1)**(1/2),x)

[Out]

x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), x**4*exp_polar(2*I*pi))/(4*gamma(13/4
))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{-x^{4} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(-x^4 + 1),x, algorithm="giac")

[Out]

integrate(x^8/sqrt(-x^4 + 1), x)