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

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 = 0.0321717, 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 = 3.50721, 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 = 0.00727513, size = 18, normalized size = 0.67 \[ \frac{2}{9} \left (x^3-2\right ) \sqrt{x^3+1} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/Sqrt[1 + x^3],x]

[Out]

(2*(-2 + x^3)*Sqrt[1 + x^3])/9

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

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 = 1.44389, 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 = 0.22043, size = 19, normalized size = 0.7 \[ \frac{2}{9} \, \sqrt{x^{3} + 1}{\left (x^{3} - 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*sqrt(x^3 + 1)*(x^3 - 2)

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Sympy [A]  time = 0.769349, 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 = 0.215379, 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)