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

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

[Out]

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

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Rubi [A]  time = 0.0402142, 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 \[ \frac{2}{9} \left (-x^3-1\right )^{3/2}+\frac{2}{3} \sqrt{-x^3-1} \]

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 = 5.10509, size = 26, normalized size = 0.84 \[ \frac{2 \left (- x^{3} - 1\right )^{\frac{3}{2}}}{9} + \frac{2 \sqrt{- x^{3} - 1}}{3} \]

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 = 0.00833844, size = 20, normalized size = 0.65 \[ -\frac{2}{9} \sqrt{-x^3-1} \left (x^3-2\right ) \]

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 = 28, normalized size = 0.9 \[{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) \left ({x}^{3}-2 \right ) }{9}{\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

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 = 1.46343, 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 = 0.226734, 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 = 0.764559, size = 29, normalized size = 0.94 \[ - \frac{2 x^{3} \sqrt{- x^{3} - 1}}{9} + \frac{4 \sqrt{- x^{3} - 1}}{9} \]

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 = 0.211061, 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)