3.1408 \(\int \frac{x^{11}}{\left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \frac{\sqrt{x^6+2}}{3}+\frac{2}{3 \sqrt{x^6+2}} \]

[Out]

2/(3*Sqrt[2 + x^6]) + Sqrt[2 + x^6]/3

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Rubi [A]  time = 0.0303273, 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 \[ \frac{\sqrt{x^6+2}}{3}+\frac{2}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^11/(2 + x^6)^(3/2),x]

[Out]

2/(3*Sqrt[2 + x^6]) + Sqrt[2 + x^6]/3

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Rubi in Sympy [A]  time = 3.67452, size = 20, normalized size = 0.74 \[ \frac{\sqrt{x^{6} + 2}}{3} + \frac{2}{3 \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(x**6+2)**(3/2),x)

[Out]

sqrt(x**6 + 2)/3 + 2/(3*sqrt(x**6 + 2))

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Mathematica [A]  time = 0.0105169, size = 18, normalized size = 0.67 \[ \frac{x^6+4}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/(2 + x^6)^(3/2),x]

[Out]

(4 + x^6)/(3*Sqrt[2 + x^6])

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Maple [A]  time = 0.006, size = 15, normalized size = 0.6 \[{\frac{{x}^{6}+4}{3}{\frac{1}{\sqrt{{x}^{6}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(x^6+2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.44236, size = 26, normalized size = 0.96 \[ \frac{1}{3} \, \sqrt{x^{6} + 2} + \frac{2}{3 \, \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^6 + 2)^(3/2),x, algorithm="maxima")

[Out]

1/3*sqrt(x^6 + 2) + 2/3/sqrt(x^6 + 2)

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Fricas [A]  time = 0.219009, size = 19, normalized size = 0.7 \[ \frac{x^{6} + 4}{3 \, \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^6 + 2)^(3/2),x, algorithm="fricas")

[Out]

1/3*(x^6 + 4)/sqrt(x^6 + 2)

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Sympy [A]  time = 5.64897, size = 24, normalized size = 0.89 \[ \frac{x^{6}}{3 \sqrt{x^{6} + 2}} + \frac{4}{3 \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(x**6+2)**(3/2),x)

[Out]

x**6/(3*sqrt(x**6 + 2)) + 4/(3*sqrt(x**6 + 2))

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GIAC/XCAS [A]  time = 0.220907, size = 26, normalized size = 0.96 \[ \frac{1}{3} \, \sqrt{x^{6} + 2} + \frac{2}{3 \, \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/(x^6 + 2)^(3/2),x, algorithm="giac")

[Out]

1/3*sqrt(x^6 + 2) + 2/3/sqrt(x^6 + 2)