3.892 \(\int \frac{x^{11}}{\left (1-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=42 \[ -\frac{1}{6} \left (1-x^4\right )^{3/2}+\sqrt{1-x^4}+\frac{1}{2 \sqrt{1-x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0528638, antiderivative size = 42, 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 \[ -\frac{1}{6} \left (1-x^4\right )^{3/2}+\sqrt{1-x^4}+\frac{1}{2 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(1 - x^4)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.99895, size = 29, normalized size = 0.69 \[ - \frac{\left (- x^{4} + 1\right )^{\frac{3}{2}}}{6} + \sqrt{- x^{4} + 1} + \frac{1}{2 \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(-x**4+1)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.016683, size = 27, normalized size = 0.64 \[ \frac{-x^8-4 x^4+8}{6 \sqrt{1-x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(1 - x^4)^(3/2),x]

[Out]

(8 - 4*x^4 - x^8)/(6*Sqrt[1 - x^4])

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Maple [A]  time = 0.008, size = 33, normalized size = 0.8 \[{\frac{ \left ( -1+x \right ) \left ( 1+x \right ) \left ({x}^{2}+1 \right ) \left ({x}^{8}+4\,{x}^{4}-8 \right ) }{6} \left ( -{x}^{4}+1 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(-x^4+1)^(3/2),x)

[Out]

1/6*(-1+x)*(1+x)*(x^2+1)*(x^8+4*x^4-8)/(-x^4+1)^(3/2)

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Maxima [A]  time = 1.43641, size = 43, normalized size = 1.02 \[ -\frac{1}{6} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} + \sqrt{-x^{4} + 1} + \frac{1}{2 \, \sqrt{-x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(-x^4 + 1)^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.282666, size = 28, normalized size = 0.67 \[ -\frac{x^{8} + 4 \, x^{4} - 8}{6 \, \sqrt{-x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(-x^4 + 1)^(3/2),x, algorithm="fricas")

[Out]

-1/6*(x^8 + 4*x^4 - 8)/sqrt(-x^4 + 1)

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Sympy [A]  time = 5.64192, size = 39, normalized size = 0.93 \[ - \frac{x^{8}}{6 \sqrt{- x^{4} + 1}} - \frac{2 x^{4}}{3 \sqrt{- x^{4} + 1}} + \frac{4}{3 \sqrt{- x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(-x**4+1)**(3/2),x)

[Out]

-x**8/(6*sqrt(-x**4 + 1)) - 2*x**4/(3*sqrt(-x**4 + 1)) + 4/(3*sqrt(-x**4 + 1))

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GIAC/XCAS [A]  time = 0.213292, size = 43, normalized size = 1.02 \[ -\frac{1}{6} \,{\left (-x^{4} + 1\right )}^{\frac{3}{2}} + \sqrt{-x^{4} + 1} + \frac{1}{2 \, \sqrt{-x^{4} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(-x^4 + 1)^(3/2),x, algorithm="giac")

[Out]

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