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

Optimal. Leaf size=27 \[ \frac{2}{9} \left (x^3-1\right )^{3/2}+\frac{2 \sqrt{x^3-1}}{3} \]

[Out]

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

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Rubi [A]  time = 0.0314489, antiderivative size = 27, 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 \[ \frac{2}{9} \left (x^3-1\right )^{3/2}+\frac{2 \sqrt{x^3-1}}{3} \]

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 = 3.30721, size = 22, normalized size = 0.81 \[ \frac{2 \left (x^{3} - 1\right )^{\frac{3}{2}}}{9} + \frac{2 \sqrt{x^{3} - 1}}{3} \]

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 = 0.00748952, size = 18, normalized size = 0.67 \[ \frac{2}{9} \sqrt{x^3-1} \left (x^3+2\right ) \]

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 = 0.007, size = 24, 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 = 1.43466, size = 26, normalized size = 0.96 \[ \frac{2}{9} \,{\left (x^{3} - 1\right )}^{\frac{3}{2}} + \frac{2}{3} \, \sqrt{x^{3} - 1} \]

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 = 0.22923, size = 19, normalized size = 0.7 \[ \frac{2}{9} \,{\left (x^{3} + 2\right )} \sqrt{x^{3} - 1} \]

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 = 0.779196, size = 26, normalized size = 0.96 \[ \frac{2 x^{3} \sqrt{x^{3} - 1}}{9} + \frac{4 \sqrt{x^{3} - 1}}{9} \]

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 = 0.216749, size = 26, normalized size = 0.96 \[ \frac{2}{9} \,{\left (x^{3} - 1\right )}^{\frac{3}{2}} + \frac{2}{3} \, \sqrt{x^{3} - 1} \]

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)