3.458 \(\int \frac{x^5}{\sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=31 \[ \frac{2}{9} \left (1-x^3\right )^{3/2}-\frac{2 \sqrt{1-x^3}}{3} \]

[Out]

(-2*Sqrt[1 - x^3])/3 + (2*(1 - x^3)^(3/2))/9

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Rubi [A]  time = 0.0410548, 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 \[ \frac{2}{9} \left (1-x^3\right )^{3/2}-\frac{2 \sqrt{1-x^3}}{3} \]

Antiderivative was successfully verified.

[In]  Int[x^5/Sqrt[1 - x^3],x]

[Out]

(-2*Sqrt[1 - x^3])/3 + (2*(1 - x^3)^(3/2))/9

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Rubi in Sympy [A]  time = 4.68414, size = 22, normalized size = 0.71 \[ \frac{2 \left (- x^{3} + 1\right )^{\frac{3}{2}}}{9} - \frac{2 \sqrt{- x^{3} + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(-x**3+1)**(1/2),x)

[Out]

2*(-x**3 + 1)**(3/2)/9 - 2*sqrt(-x**3 + 1)/3

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Mathematica [A]  time = 0.00845523, size = 20, normalized size = 0.65 \[ -\frac{2}{9} \sqrt{1-x^3} \left (x^3+2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/Sqrt[1 - x^3],x]

[Out]

(-2*Sqrt[1 - x^3]*(2 + x^3))/9

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Maple [A]  time = 0.006, size = 26, normalized size = 0.8 \[{\frac{ \left ( -2+2\,x \right ) \left ({x}^{2}+x+1 \right ) \left ({x}^{3}+2 \right ) }{9}{\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(-x^3+1)^(1/2),x)

[Out]

2/9*(-1+x)*(x^2+x+1)*(x^3+2)/(-x^3+1)^(1/2)

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Maxima [A]  time = 1.42288, size = 31, normalized size = 1. \[ \frac{2}{9} \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{-x^{3} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-x^3 + 1),x, algorithm="maxima")

[Out]

2/9*(-x^3 + 1)^(3/2) - 2/3*sqrt(-x^3 + 1)

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Fricas [A]  time = 0.227605, size = 22, normalized size = 0.71 \[ -\frac{2}{9} \,{\left (x^{3} + 2\right )} \sqrt{-x^{3} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-x^3 + 1),x, algorithm="fricas")

[Out]

-2/9*(x^3 + 2)*sqrt(-x^3 + 1)

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Sympy [A]  time = 0.780375, size = 27, normalized size = 0.87 \[ - \frac{2 x^{3} \sqrt{- x^{3} + 1}}{9} - \frac{4 \sqrt{- x^{3} + 1}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(-x**3+1)**(1/2),x)

[Out]

-2*x**3*sqrt(-x**3 + 1)/9 - 4*sqrt(-x**3 + 1)/9

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GIAC/XCAS [A]  time = 0.218833, size = 31, normalized size = 1. \[ \frac{2}{9} \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} - \frac{2}{3} \, \sqrt{-x^{3} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/sqrt(-x^3 + 1),x, algorithm="giac")

[Out]

2/9*(-x^3 + 1)^(3/2) - 2/3*sqrt(-x^3 + 1)