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

Optimal. Leaf size=53 \[ \frac{2}{21} \left (x^3-1\right )^{7/2}+\frac{2}{5} \left (x^3-1\right )^{5/2}+\frac{2}{3} \left (x^3-1\right )^{3/2}+\frac{2 \sqrt{x^3-1}}{3} \]

[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 = 0.0489584, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2}{21} \left (x^3-1\right )^{7/2}+\frac{2}{5} \left (x^3-1\right )^{5/2}+\frac{2}{3} \left (x^3-1\right )^{3/2}+\frac{2 \sqrt{x^3-1}}{3} \]

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 = 4.77472, size = 46, 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*sqrt(
x**3 - 1)/3

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Mathematica [A]  time = 0.014509, size = 30, normalized size = 0.57 \[ \frac{2}{105} \sqrt{x^3-1} \left (5 x^9+6 x^6+8 x^3+16\right ) \]

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 = 36, 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 = 1.44141, size = 50, normalized size = 0.94 \[ \frac{2}{21} \,{\left (x^{3} - 1\right )}^{\frac{7}{2}} + \frac{2}{5} \,{\left (x^{3} - 1\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (x^{3} - 1\right )}^{\frac{3}{2}} + \frac{2}{3} \, \sqrt{x^{3} - 1} \]

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 = 0.230948, size = 35, normalized size = 0.66 \[ \frac{2}{105} \,{\left (5 \, x^{9} + 6 \, x^{6} + 8 \, x^{3} + 16\right )} \sqrt{x^{3} - 1} \]

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 = 4.54106, size = 56, normalized size = 1.06 \[ \frac{2 x^{9} \sqrt{x^{3} - 1}}{21} + \frac{4 x^{6} \sqrt{x^{3} - 1}}{35} + \frac{16 x^{3} \sqrt{x^{3} - 1}}{105} + \frac{32 \sqrt{x^{3} - 1}}{105} \]

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 = 0.221385, size = 50, normalized size = 0.94 \[ \frac{2}{21} \,{\left (x^{3} - 1\right )}^{\frac{7}{2}} + \frac{2}{5} \,{\left (x^{3} - 1\right )}^{\frac{5}{2}} + \frac{2}{3} \,{\left (x^{3} - 1\right )}^{\frac{3}{2}} + \frac{2}{3} \, \sqrt{x^{3} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[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)