3.873 \(\int \frac{x^7}{\sqrt{1-x^4}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{6} \left (1-x^4\right )^{3/2}-\frac{\sqrt{1-x^4}}{2} \]

[Out]

-Sqrt[1 - x^4]/2 + (1 - x^4)^(3/2)/6

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Rubi [A]  time = 0.0403579, antiderivative size = 31, 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 \[ \frac{1}{6} \left (1-x^4\right )^{3/2}-\frac{\sqrt{1-x^4}}{2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/Sqrt[1 - x^4],x]

[Out]

-Sqrt[1 - x^4]/2 + (1 - x^4)^(3/2)/6

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Rubi in Sympy [A]  time = 4.56117, size = 19, normalized size = 0.61 \[ \frac{\left (- x^{4} + 1\right )^{\frac{3}{2}}}{6} - \frac{\sqrt{- x^{4} + 1}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(-x**4+1)**(1/2),x)

[Out]

(-x**4 + 1)**(3/2)/6 - sqrt(-x**4 + 1)/2

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Mathematica [A]  time = 0.0098814, size = 20, normalized size = 0.65 \[ -\frac{1}{6} \sqrt{1-x^4} \left (x^4+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/Sqrt[1 - x^4],x]

[Out]

-(Sqrt[1 - x^4]*(2 + x^4))/6

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Maple [A]  time = 0.006, size = 28, normalized size = 0.9 \[{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ({x}^{4}+2 \right ) }{6}{\frac{1}{\sqrt{-{x}^{4}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(-x^4+1)^(1/2),x)

[Out]

1/6*(-1+x)*(1+x)*(x^2+1)*(x^4+2)/(-x^4+1)^(1/2)

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Maxima [A]  time = 1.43179, size = 31, normalized size = 1. \[ \frac{1}{6} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(-x^4 + 1),x, algorithm="maxima")

[Out]

1/6*(-x^4 + 1)^(3/2) - 1/2*sqrt(-x^4 + 1)

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Fricas [A]  time = 0.248508, size = 22, normalized size = 0.71 \[ -\frac{1}{6} \,{\left (x^{4} + 2\right )} \sqrt{-x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(-x^4 + 1),x, algorithm="fricas")

[Out]

-1/6*(x^4 + 2)*sqrt(-x^4 + 1)

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Sympy [A]  time = 1.31124, size = 24, normalized size = 0.77 \[ - \frac{x^{4} \sqrt{- x^{4} + 1}}{6} - \frac{\sqrt{- x^{4} + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(-x**4+1)**(1/2),x)

[Out]

-x**4*sqrt(-x**4 + 1)/6 - sqrt(-x**4 + 1)/3

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GIAC/XCAS [A]  time = 0.210485, size = 31, normalized size = 1. \[ \frac{1}{6} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} - \frac{1}{2} \, \sqrt{-x^{4} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/sqrt(-x^4 + 1),x, algorithm="giac")

[Out]

1/6*(-x^4 + 1)^(3/2) - 1/2*sqrt(-x^4 + 1)