3.1406 \(\int \frac{x^{23}}{\left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=51 \[ \frac{1}{15} \left (x^6+2\right )^{5/2}-\frac{2}{3} \left (x^6+2\right )^{3/2}+4 \sqrt{x^6+2}+\frac{8}{3 \sqrt{x^6+2}} \]

[Out]

8/(3*Sqrt[2 + x^6]) + 4*Sqrt[2 + x^6] - (2*(2 + x^6)^(3/2))/3 + (2 + x^6)^(5/2)/
15

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Rubi [A]  time = 0.0461454, antiderivative size = 51, 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 \[ \frac{1}{15} \left (x^6+2\right )^{5/2}-\frac{2}{3} \left (x^6+2\right )^{3/2}+4 \sqrt{x^6+2}+\frac{8}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^23/(2 + x^6)^(3/2),x]

[Out]

8/(3*Sqrt[2 + x^6]) + 4*Sqrt[2 + x^6] - (2*(2 + x^6)^(3/2))/3 + (2 + x^6)^(5/2)/
15

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Rubi in Sympy [A]  time = 5.1518, size = 42, normalized size = 0.82 \[ \frac{\left (x^{6} + 2\right )^{\frac{5}{2}}}{15} - \frac{2 \left (x^{6} + 2\right )^{\frac{3}{2}}}{3} + 4 \sqrt{x^{6} + 2} + \frac{8}{3 \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**23/(x**6+2)**(3/2),x)

[Out]

(x**6 + 2)**(5/2)/15 - 2*(x**6 + 2)**(3/2)/3 + 4*sqrt(x**6 + 2) + 8/(3*sqrt(x**6
+ 2))

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Mathematica [A]  time = 0.017889, size = 28, normalized size = 0.55 \[ \frac{x^{18}-4 x^{12}+32 x^6+128}{15 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^23/(2 + x^6)^(3/2),x]

[Out]

(128 + 32*x^6 - 4*x^12 + x^18)/(15*Sqrt[2 + x^6])

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Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[{\frac{{x}^{18}-4\,{x}^{12}+32\,{x}^{6}+128}{15}{\frac{1}{\sqrt{{x}^{6}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^23/(x^6+2)^(3/2),x)

[Out]

1/15*(x^18-4*x^12+32*x^6+128)/(x^6+2)^(1/2)

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Maxima [A]  time = 1.44514, size = 50, normalized size = 0.98 \[ \frac{1}{15} \,{\left (x^{6} + 2\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} + 4 \, \sqrt{x^{6} + 2} + \frac{8}{3 \, \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^23/(x^6 + 2)^(3/2),x, algorithm="maxima")

[Out]

1/15*(x^6 + 2)^(5/2) - 2/3*(x^6 + 2)^(3/2) + 4*sqrt(x^6 + 2) + 8/3/sqrt(x^6 + 2)

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Fricas [A]  time = 0.216945, size = 32, normalized size = 0.63 \[ \frac{x^{18} - 4 \, x^{12} + 32 \, x^{6} + 128}{15 \, \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^23/(x^6 + 2)^(3/2),x, algorithm="fricas")

[Out]

1/15*(x^18 - 4*x^12 + 32*x^6 + 128)/sqrt(x^6 + 2)

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Sympy [A]  time = 47.1398, size = 54, normalized size = 1.06 \[ \frac{x^{18}}{15 \sqrt{x^{6} + 2}} - \frac{4 x^{12}}{15 \sqrt{x^{6} + 2}} + \frac{32 x^{6}}{15 \sqrt{x^{6} + 2}} + \frac{128}{15 \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**23/(x**6+2)**(3/2),x)

[Out]

x**18/(15*sqrt(x**6 + 2)) - 4*x**12/(15*sqrt(x**6 + 2)) + 32*x**6/(15*sqrt(x**6
+ 2)) + 128/(15*sqrt(x**6 + 2))

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GIAC/XCAS [A]  time = 0.219925, size = 50, normalized size = 0.98 \[ \frac{1}{15} \,{\left (x^{6} + 2\right )}^{\frac{5}{2}} - \frac{2}{3} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} + 4 \, \sqrt{x^{6} + 2} + \frac{8}{3 \, \sqrt{x^{6} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^23/(x^6 + 2)^(3/2),x, algorithm="giac")

[Out]

1/15*(x^6 + 2)^(5/2) - 2/3*(x^6 + 2)^(3/2) + 4*sqrt(x^6 + 2) + 8/3/sqrt(x^6 + 2)