3.439 \(\int \frac{x^8}{\sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=40 \[ \frac{2}{15} \left (x^3+1\right )^{5/2}-\frac{4}{9} \left (x^3+1\right )^{3/2}+\frac{2 \sqrt{x^3+1}}{3} \]

[Out]

(2*Sqrt[1 + x^3])/3 - (4*(1 + x^3)^(3/2))/9 + (2*(1 + x^3)^(5/2))/15

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Rubi [A]  time = 0.0423354, antiderivative size = 40, 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 \[ \frac{2}{15} \left (x^3+1\right )^{5/2}-\frac{4}{9} \left (x^3+1\right )^{3/2}+\frac{2 \sqrt{x^3+1}}{3} \]

Antiderivative was successfully verified.

[In]  Int[x^8/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[1 + x^3])/3 - (4*(1 + x^3)^(3/2))/9 + (2*(1 + x^3)^(5/2))/15

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Rubi in Sympy [A]  time = 4.17593, size = 34, normalized size = 0.85 \[ \frac{2 \left (x^{3} + 1\right )^{\frac{5}{2}}}{15} - \frac{4 \left (x^{3} + 1\right )^{\frac{3}{2}}}{9} + \frac{2 \sqrt{x^{3} + 1}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(x**3+1)**(1/2),x)

[Out]

2*(x**3 + 1)**(5/2)/15 - 4*(x**3 + 1)**(3/2)/9 + 2*sqrt(x**3 + 1)/3

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Mathematica [A]  time = 0.0130601, size = 25, normalized size = 0.62 \[ \frac{2}{45} \sqrt{x^3+1} \left (3 x^6-4 x^3+8\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/Sqrt[1 + x^3],x]

[Out]

(2*Sqrt[1 + x^3]*(8 - 4*x^3 + 3*x^6))/45

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Maple [A]  time = 0.006, size = 33, normalized size = 0.8 \[{\frac{ \left ( 2+2\,x \right ) \left ({x}^{2}-x+1 \right ) \left ( 3\,{x}^{6}-4\,{x}^{3}+8 \right ) }{45}{\frac{1}{\sqrt{{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(x^3+1)^(1/2),x)

[Out]

2/45*(1+x)*(x^2-x+1)*(3*x^6-4*x^3+8)/(x^3+1)^(1/2)

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Maxima [A]  time = 1.43205, size = 38, normalized size = 0.95 \[ \frac{2}{15} \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} - \frac{4}{9} \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} + \frac{2}{3} \, \sqrt{x^{3} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(x^3 + 1),x, algorithm="maxima")

[Out]

2/15*(x^3 + 1)^(5/2) - 4/9*(x^3 + 1)^(3/2) + 2/3*sqrt(x^3 + 1)

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Fricas [A]  time = 0.220791, size = 28, normalized size = 0.7 \[ \frac{2}{45} \,{\left (3 \, x^{6} - 4 \, x^{3} + 8\right )} \sqrt{x^{3} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(x^3 + 1),x, algorithm="fricas")

[Out]

2/45*(3*x^6 - 4*x^3 + 8)*sqrt(x^3 + 1)

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Sympy [A]  time = 1.91906, size = 41, normalized size = 1.02 \[ \frac{2 x^{6} \sqrt{x^{3} + 1}}{15} - \frac{8 x^{3} \sqrt{x^{3} + 1}}{45} + \frac{16 \sqrt{x^{3} + 1}}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(x**3+1)**(1/2),x)

[Out]

2*x**6*sqrt(x**3 + 1)/15 - 8*x**3*sqrt(x**3 + 1)/45 + 16*sqrt(x**3 + 1)/45

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GIAC/XCAS [A]  time = 0.223494, size = 38, normalized size = 0.95 \[ \frac{2}{15} \,{\left (x^{3} + 1\right )}^{\frac{5}{2}} - \frac{4}{9} \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} + \frac{2}{3} \, \sqrt{x^{3} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(x^3 + 1),x, algorithm="giac")

[Out]

2/15*(x^3 + 1)^(5/2) - 4/9*(x^3 + 1)^(3/2) + 2/3*sqrt(x^3 + 1)