3.1393 \(\int \frac{1}{x^{16} \sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=49 \[ -\frac{\sqrt{x^6+2}}{30 x^{15}}+\frac{\sqrt{x^6+2}}{45 x^9}-\frac{\sqrt{x^6+2}}{45 x^3} \]

[Out]

-Sqrt[2 + x^6]/(30*x^15) + Sqrt[2 + x^6]/(45*x^9) - Sqrt[2 + x^6]/(45*x^3)

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Rubi [A]  time = 0.0382607, antiderivative size = 49, 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 \[ -\frac{\sqrt{x^6+2}}{30 x^{15}}+\frac{\sqrt{x^6+2}}{45 x^9}-\frac{\sqrt{x^6+2}}{45 x^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^16*Sqrt[2 + x^6]),x]

[Out]

-Sqrt[2 + x^6]/(30*x^15) + Sqrt[2 + x^6]/(45*x^9) - Sqrt[2 + x^6]/(45*x^3)

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Rubi in Sympy [A]  time = 4.31169, size = 39, normalized size = 0.8 \[ - \frac{\sqrt{x^{6} + 2}}{45 x^{3}} + \frac{\sqrt{x^{6} + 2}}{45 x^{9}} - \frac{\sqrt{x^{6} + 2}}{30 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**16/(x**6+2)**(1/2),x)

[Out]

-sqrt(x**6 + 2)/(45*x**3) + sqrt(x**6 + 2)/(45*x**9) - sqrt(x**6 + 2)/(30*x**15)

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Mathematica [A]  time = 0.01534, size = 28, normalized size = 0.57 \[ -\frac{\sqrt{x^6+2} \left (2 x^{12}-2 x^6+3\right )}{90 x^{15}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^16*Sqrt[2 + x^6]),x]

[Out]

-(Sqrt[2 + x^6]*(3 - 2*x^6 + 2*x^12))/(90*x^15)

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Maple [A]  time = 0.006, size = 25, normalized size = 0.5 \[ -{\frac{2\,{x}^{12}-2\,{x}^{6}+3}{90\,{x}^{15}}\sqrt{{x}^{6}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^16/(x^6+2)^(1/2),x)

[Out]

-1/90*(x^6+2)^(1/2)*(2*x^12-2*x^6+3)/x^15

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Maxima [A]  time = 1.43357, size = 50, normalized size = 1.02 \[ -\frac{\sqrt{x^{6} + 2}}{24 \, x^{3}} + \frac{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}{36 \, x^{9}} - \frac{{\left (x^{6} + 2\right )}^{\frac{5}{2}}}{120 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^16),x, algorithm="maxima")

[Out]

-1/24*sqrt(x^6 + 2)/x^3 + 1/36*(x^6 + 2)^(3/2)/x^9 - 1/120*(x^6 + 2)^(5/2)/x^15

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Fricas [A]  time = 0.219294, size = 101, normalized size = 2.06 \[ \frac{20 \, x^{12} + 35 \, x^{6} - 5 \,{\left (4 \, x^{9} + 3 \, x^{3}\right )} \sqrt{x^{6} + 2} + 6}{90 \,{\left (4 \, x^{30} + 10 \, x^{24} + 5 \, x^{18} -{\left (4 \, x^{27} + 6 \, x^{21} + x^{15}\right )} \sqrt{x^{6} + 2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^16),x, algorithm="fricas")

[Out]

1/90*(20*x^12 + 35*x^6 - 5*(4*x^9 + 3*x^3)*sqrt(x^6 + 2) + 6)/(4*x^30 + 10*x^24
+ 5*x^18 - (4*x^27 + 6*x^21 + x^15)*sqrt(x^6 + 2))

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Sympy [A]  time = 16.5713, size = 41, normalized size = 0.84 \[ - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{45} + \frac{\sqrt{1 + \frac{2}{x^{6}}}}{45 x^{6}} - \frac{\sqrt{1 + \frac{2}{x^{6}}}}{30 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**16/(x**6+2)**(1/2),x)

[Out]

-sqrt(1 + 2/x**6)/45 + sqrt(1 + 2/x**6)/(45*x**6) - sqrt(1 + 2/x**6)/(30*x**12)

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GIAC/XCAS [A]  time = 0.226399, size = 61, normalized size = 1.24 \[ -\frac{3 \,{\left (\frac{2}{x^{6}} + 1\right )}^{\frac{5}{2}} - 10 \,{\left (\frac{2}{x^{6}} + 1\right )}^{\frac{3}{2}} + 15 \, \sqrt{\frac{2}{x^{6}} + 1}}{360 \,{\rm sign}\left (x\right )} + \frac{1}{45} \,{\rm sign}\left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^6 + 2)*x^16),x, algorithm="giac")

[Out]

-1/360*(3*(2/x^6 + 1)^(5/2) - 10*(2/x^6 + 1)^(3/2) + 15*sqrt(2/x^6 + 1))/sign(x)
+ 1/45*sign(x)