3.492 \(\int \frac{x^{11}}{\sqrt{-1-x^3}} \, dx\)

Optimal. Leaf size=61 \[ \frac{2}{21} \left (-x^3-1\right )^{7/2}+\frac{2}{5} \left (-x^3-1\right )^{5/2}+\frac{2}{3} \left (-x^3-1\right )^{3/2}+\frac{2}{3} \sqrt{-x^3-1} \]

[Out]

(2*Sqrt[-1 - x^3])/3 + (2*(-1 - x^3)^(3/2))/3 + (2*(-1 - x^3)^(5/2))/5 + (2*(-1
- x^3)^(7/2))/21

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Rubi [A]  time = 0.0659667, antiderivative size = 61, 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 \[ \frac{2}{21} \left (-x^3-1\right )^{7/2}+\frac{2}{5} \left (-x^3-1\right )^{5/2}+\frac{2}{3} \left (-x^3-1\right )^{3/2}+\frac{2}{3} \sqrt{-x^3-1} \]

Antiderivative was successfully verified.

[In]  Int[x^11/Sqrt[-1 - x^3],x]

[Out]

(2*Sqrt[-1 - x^3])/3 + (2*(-1 - x^3)^(3/2))/3 + (2*(-1 - x^3)^(5/2))/5 + (2*(-1
- x^3)^(7/2))/21

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Rubi in Sympy [A]  time = 7.32734, size = 53, normalized size = 0.87 \[ \frac{2 \left (- x^{3} - 1\right )^{\frac{7}{2}}}{21} + \frac{2 \left (- x^{3} - 1\right )^{\frac{5}{2}}}{5} + \frac{2 \left (- x^{3} - 1\right )^{\frac{3}{2}}}{3} + \frac{2 \sqrt{- x^{3} - 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-x**3 - 1)**(7/2)/21 + 2*(-x**3 - 1)**(5/2)/5 + 2*(-x**3 - 1)**(3/2)/3 + 2*sq
rt(-x**3 - 1)/3

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Mathematica [A]  time = 0.0167002, size = 32, normalized size = 0.52 \[ -\frac{2}{105} \sqrt{-x^3-1} \left (5 x^9-6 x^6+8 x^3-16\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/Sqrt[-1 - x^3],x]

[Out]

(-2*Sqrt[-1 - x^3]*(-16 + 8*x^3 - 6*x^6 + 5*x^9))/105

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Maple [A]  time = 0.007, size = 40, normalized size = 0.7 \[{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) \left ( 5\,{x}^{9}-6\,{x}^{6}+8\,{x}^{3}-16 \right ) }{105}{\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(-x^3-1)^(1/2),x)

[Out]

2/105*(1+x)*(x^2-x+1)*(5*x^9-6*x^6+8*x^3-16)/(-x^3-1)^(1/2)

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Maxima [A]  time = 1.43992, size = 61, normalized size = 1. \[ \frac{2}{21} \,{\left (-x^{3} - 1\right )}^{\frac{7}{2}} + \frac{2}{5} \,{\left (-x^{3} - 1\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (-x^{3} - 1\right )}^{\frac{3}{2}} + \frac{2}{3} \, \sqrt{-x^{3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/21*(-x^3 - 1)^(7/2) + 2/5*(-x^3 - 1)^(5/2) + 2/3*(-x^3 - 1)^(3/2) + 2/3*sqrt(-
x^3 - 1)

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Fricas [A]  time = 0.225686, size = 38, normalized size = 0.62 \[ -\frac{2}{105} \,{\left (5 \, x^{9} - 6 \, x^{6} + 8 \, x^{3} - 16\right )} \sqrt{-x^{3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/105*(5*x^9 - 6*x^6 + 8*x^3 - 16)*sqrt(-x^3 - 1)

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Sympy [A]  time = 4.50879, size = 63, normalized size = 1.03 \[ - \frac{2 x^{9} \sqrt{- x^{3} - 1}}{21} + \frac{4 x^{6} \sqrt{- x^{3} - 1}}{35} - \frac{16 x^{3} \sqrt{- x^{3} - 1}}{105} + \frac{32 \sqrt{- x^{3} - 1}}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**9*sqrt(-x**3 - 1)/21 + 4*x**6*sqrt(-x**3 - 1)/35 - 16*x**3*sqrt(-x**3 - 1)
/105 + 32*sqrt(-x**3 - 1)/105

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GIAC/XCAS [A]  time = 0.212613, size = 80, normalized size = 1.31 \[ -\frac{2}{21} \,{\left (x^{3} + 1\right )}^{3} \sqrt{-x^{3} - 1} + \frac{2}{5} \,{\left (x^{3} + 1\right )}^{2} \sqrt{-x^{3} - 1} + \frac{2}{3} \,{\left (-x^{3} - 1\right )}^{\frac{3}{2}} + \frac{2}{3} \, \sqrt{-x^{3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/21*(x^3 + 1)^3*sqrt(-x^3 - 1) + 2/5*(x^3 + 1)^2*sqrt(-x^3 - 1) + 2/3*(-x^3 -
1)^(3/2) + 2/3*sqrt(-x^3 - 1)