3.1382 \(\int \frac{x^{17}}{\sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=40 \[ \frac{1}{15} \left (x^6+2\right )^{5/2}-\frac{4}{9} \left (x^6+2\right )^{3/2}+\frac{4 \sqrt{x^6+2}}{3} \]

[Out]

(4*Sqrt[2 + x^6])/3 - (4*(2 + x^6)^(3/2))/9 + (2 + x^6)^(5/2)/15

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Rubi [A]  time = 0.0389995, 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 \[ \frac{1}{15} \left (x^6+2\right )^{5/2}-\frac{4}{9} \left (x^6+2\right )^{3/2}+\frac{4 \sqrt{x^6+2}}{3} \]

Antiderivative was successfully verified.

[In]  Int[x^17/Sqrt[2 + x^6],x]

[Out]

(4*Sqrt[2 + x^6])/3 - (4*(2 + x^6)^(3/2))/9 + (2 + x^6)^(5/2)/15

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Rubi in Sympy [A]  time = 4.47041, size = 32, normalized size = 0.8 \[ \frac{\left (x^{6} + 2\right )^{\frac{5}{2}}}{15} - \frac{4 \left (x^{6} + 2\right )^{\frac{3}{2}}}{9} + \frac{4 \sqrt{x^{6} + 2}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**17/(x**6+2)**(1/2),x)

[Out]

(x**6 + 2)**(5/2)/15 - 4*(x**6 + 2)**(3/2)/9 + 4*sqrt(x**6 + 2)/3

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Mathematica [A]  time = 0.0132902, size = 25, normalized size = 0.62 \[ \frac{1}{45} \sqrt{x^6+2} \left (3 x^{12}-8 x^6+32\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^17/Sqrt[2 + x^6],x]

[Out]

(Sqrt[2 + x^6]*(32 - 8*x^6 + 3*x^12))/45

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Maple [A]  time = 0.006, size = 22, normalized size = 0.6 \[{\frac{3\,{x}^{12}-8\,{x}^{6}+32}{45}\sqrt{{x}^{6}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^17/(x^6+2)^(1/2),x)

[Out]

1/45*(x^6+2)^(1/2)*(3*x^12-8*x^6+32)

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Maxima [A]  time = 1.41831, size = 38, normalized size = 0.95 \[ \frac{1}{15} \,{\left (x^{6} + 2\right )}^{\frac{5}{2}} - \frac{4}{9} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} + \frac{4}{3} \, \sqrt{x^{6} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^17/sqrt(x^6 + 2),x, algorithm="maxima")

[Out]

1/15*(x^6 + 2)^(5/2) - 4/9*(x^6 + 2)^(3/2) + 4/3*sqrt(x^6 + 2)

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Fricas [A]  time = 0.22035, size = 28, normalized size = 0.7 \[ \frac{1}{45} \,{\left (3 \, x^{12} - 8 \, x^{6} + 32\right )} \sqrt{x^{6} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^17/sqrt(x^6 + 2),x, algorithm="fricas")

[Out]

1/45*(3*x^12 - 8*x^6 + 32)*sqrt(x^6 + 2)

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Sympy [A]  time = 15.8572, size = 39, normalized size = 0.98 \[ \frac{x^{12} \sqrt{x^{6} + 2}}{15} - \frac{8 x^{6} \sqrt{x^{6} + 2}}{45} + \frac{32 \sqrt{x^{6} + 2}}{45} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**17/(x**6+2)**(1/2),x)

[Out]

x**12*sqrt(x**6 + 2)/15 - 8*x**6*sqrt(x**6 + 2)/45 + 32*sqrt(x**6 + 2)/45

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GIAC/XCAS [A]  time = 0.220276, size = 38, normalized size = 0.95 \[ \frac{1}{15} \,{\left (x^{6} + 2\right )}^{\frac{5}{2}} - \frac{4}{9} \,{\left (x^{6} + 2\right )}^{\frac{3}{2}} + \frac{4}{3} \, \sqrt{x^{6} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^17/sqrt(x^6 + 2),x, algorithm="giac")

[Out]

1/15*(x^6 + 2)^(5/2) - 4/9*(x^6 + 2)^(3/2) + 4/3*sqrt(x^6 + 2)