3.1381 \(\int \frac{x^{23}}{\sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=53 \[ \frac{1}{21} \left (x^6+2\right )^{7/2}-\frac{2}{5} \left (x^6+2\right )^{5/2}+\frac{4}{3} \left (x^6+2\right )^{3/2}-\frac{8 \sqrt{x^6+2}}{3} \]

[Out]

(-8*Sqrt[2 + x^6])/3 + (4*(2 + x^6)^(3/2))/3 - (2*(2 + x^6)^(5/2))/5 + (2 + x^6)
^(7/2)/21

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Rubi [A]  time = 0.046026, antiderivative size = 53, 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 \[ \frac{1}{21} \left (x^6+2\right )^{7/2}-\frac{2}{5} \left (x^6+2\right )^{5/2}+\frac{4}{3} \left (x^6+2\right )^{3/2}-\frac{8 \sqrt{x^6+2}}{3} \]

Antiderivative was successfully verified.

[In]  Int[x^23/Sqrt[2 + x^6],x]

[Out]

(-8*Sqrt[2 + x^6])/3 + (4*(2 + x^6)^(3/2))/3 - (2*(2 + x^6)^(5/2))/5 + (2 + x^6)
^(7/2)/21

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Rubi in Sympy [A]  time = 5.13452, size = 44, normalized size = 0.83 \[ \frac{\left (x^{6} + 2\right )^{\frac{7}{2}}}{21} - \frac{2 \left (x^{6} + 2\right )^{\frac{5}{2}}}{5} + \frac{4 \left (x^{6} + 2\right )^{\frac{3}{2}}}{3} - \frac{8 \sqrt{x^{6} + 2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**23/(x**6+2)**(1/2),x)

[Out]

(x**6 + 2)**(7/2)/21 - 2*(x**6 + 2)**(5/2)/5 + 4*(x**6 + 2)**(3/2)/3 - 8*sqrt(x*
*6 + 2)/3

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Mathematica [A]  time = 0.0157796, size = 30, normalized size = 0.57 \[ \frac{1}{105} \sqrt{x^6+2} \left (5 x^{18}-12 x^{12}+32 x^6-128\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^23/Sqrt[2 + x^6],x]

[Out]

(Sqrt[2 + x^6]*(-128 + 32*x^6 - 12*x^12 + 5*x^18))/105

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Maple [A]  time = 0.01, size = 27, normalized size = 0.5 \[{\frac{5\,{x}^{18}-12\,{x}^{12}+32\,{x}^{6}-128}{105}\sqrt{{x}^{6}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^23/(x^6+2)^(1/2),x)

[Out]

1/105*(x^6+2)^(1/2)*(5*x^18-12*x^12+32*x^6-128)

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Maxima [A]  time = 1.49822, size = 50, normalized size = 0.94 \[ \frac{1}{21} \,{\left (x^{6} + 2\right )}^{\frac{7}{2}} - \frac{2}{5} \,{\left (x^{6} + 2\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - \frac{8}{3} \, \sqrt{x^{6} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*(x^6 + 2)^(7/2) - 2/5*(x^6 + 2)^(5/2) + 4/3*(x^6 + 2)^(3/2) - 8/3*sqrt(x^6
+ 2)

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Fricas [A]  time = 0.221083, size = 35, normalized size = 0.66 \[ \frac{1}{105} \,{\left (5 \, x^{18} - 12 \, x^{12} + 32 \, x^{6} - 128\right )} \sqrt{x^{6} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(5*x^18 - 12*x^12 + 32*x^6 - 128)*sqrt(x^6 + 2)

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Sympy [A]  time = 40.445, size = 54, normalized size = 1.02 \[ \frac{x^{18} \sqrt{x^{6} + 2}}{21} - \frac{4 x^{12} \sqrt{x^{6} + 2}}{35} + \frac{32 x^{6} \sqrt{x^{6} + 2}}{105} - \frac{128 \sqrt{x^{6} + 2}}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**23/(x**6+2)**(1/2),x)

[Out]

x**18*sqrt(x**6 + 2)/21 - 4*x**12*sqrt(x**6 + 2)/35 + 32*x**6*sqrt(x**6 + 2)/105
- 128*sqrt(x**6 + 2)/105

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GIAC/XCAS [A]  time = 0.218453, size = 50, normalized size = 0.94 \[ \frac{1}{21} \,{\left (x^{6} + 2\right )}^{\frac{7}{2}} - \frac{2}{5} \,{\left (x^{6} + 2\right )}^{\frac{5}{2}} + \frac{4}{3} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} - \frac{8}{3} \, \sqrt{x^{6} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/21*(x^6 + 2)^(7/2) - 2/5*(x^6 + 2)^(5/2) + 4/3*(x^6 + 2)^(3/2) - 8/3*sqrt(x^6
+ 2)