3.1388 \(\int \frac{x^{14}}{\sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{2} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )+\frac{1}{12} \sqrt{x^6+2} x^9-\frac{1}{4} \sqrt{x^6+2} x^3 \]

[Out]

-(x^3*Sqrt[2 + x^6])/4 + (x^9*Sqrt[2 + x^6])/12 + ArcSinh[x^3/Sqrt[2]]/2

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Rubi [A]  time = 0.0549766, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1}{2} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )+\frac{1}{12} \sqrt{x^6+2} x^9-\frac{1}{4} \sqrt{x^6+2} x^3 \]

Antiderivative was successfully verified.

[In]  Int[x^14/Sqrt[2 + x^6],x]

[Out]

-(x^3*Sqrt[2 + x^6])/4 + (x^9*Sqrt[2 + x^6])/12 + ArcSinh[x^3/Sqrt[2]]/2

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Rubi in Sympy [A]  time = 6.30844, size = 39, normalized size = 0.83 \[ \frac{x^{9} \sqrt{x^{6} + 2}}{12} - \frac{x^{3} \sqrt{x^{6} + 2}}{4} + \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(x**6+2)**(1/2),x)

[Out]

x**9*sqrt(x**6 + 2)/12 - x**3*sqrt(x**6 + 2)/4 + asinh(sqrt(2)*x**3/2)/2

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Mathematica [A]  time = 0.0320508, size = 35, normalized size = 0.74 \[ \frac{1}{12} \left (6 \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )+\left (x^6-3\right ) \sqrt{x^6+2} x^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/Sqrt[2 + x^6],x]

[Out]

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

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Maple [A]  time = 0.045, size = 30, normalized size = 0.6 \[{\frac{{x}^{3} \left ({x}^{6}-3 \right ) }{12}\sqrt{{x}^{6}+2}}+{\frac{1}{2}{\it Arcsinh} \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(x^6+2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43316, size = 116, normalized size = 2.47 \[ -\frac{\frac{5 \, \sqrt{x^{6} + 2}}{x^{3}} - \frac{3 \,{\left (x^{6} + 2\right )}^{\frac{3}{2}}}{x^{9}}}{6 \,{\left (\frac{2 \,{\left (x^{6} + 2\right )}}{x^{6}} - \frac{{\left (x^{6} + 2\right )}^{2}}{x^{12}} - 1\right )}} + \frac{1}{4} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} + 1\right ) - \frac{1}{4} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/sqrt(x^6 + 2),x, algorithm="maxima")

[Out]

-1/6*(5*sqrt(x^6 + 2)/x^3 - 3*(x^6 + 2)^(3/2)/x^9)/(2*(x^6 + 2)/x^6 - (x^6 + 2)^
2/x^12 - 1) + 1/4*log(sqrt(x^6 + 2)/x^3 + 1) - 1/4*log(sqrt(x^6 + 2)/x^3 - 1)

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Fricas [A]  time = 0.226304, size = 165, normalized size = 3.51 \[ -\frac{2 \, x^{24} - 14 \, x^{12} - 12 \, x^{6} + 6 \,{\left (2 \, x^{12} + 4 \, x^{6} - 2 \,{\left (x^{9} + x^{3}\right )} \sqrt{x^{6} + 2} + 1\right )} \log \left (-x^{3} + \sqrt{x^{6} + 2}\right ) -{\left (2 \, x^{21} - 2 \, x^{15} - 11 \, x^{9} - 3 \, x^{3}\right )} \sqrt{x^{6} + 2}}{12 \,{\left (2 \, x^{12} + 4 \, x^{6} - 2 \,{\left (x^{9} + x^{3}\right )} \sqrt{x^{6} + 2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/sqrt(x^6 + 2),x, algorithm="fricas")

[Out]

-1/12*(2*x^24 - 14*x^12 - 12*x^6 + 6*(2*x^12 + 4*x^6 - 2*(x^9 + x^3)*sqrt(x^6 +
2) + 1)*log(-x^3 + sqrt(x^6 + 2)) - (2*x^21 - 2*x^15 - 11*x^9 - 3*x^3)*sqrt(x^6
+ 2))/(2*x^12 + 4*x^6 - 2*(x^9 + x^3)*sqrt(x^6 + 2) + 1)

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Sympy [A]  time = 14.8611, size = 53, normalized size = 1.13 \[ \frac{x^{15}}{12 \sqrt{x^{6} + 2}} - \frac{x^{9}}{12 \sqrt{x^{6} + 2}} - \frac{x^{3}}{2 \sqrt{x^{6} + 2}} + \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(x**6+2)**(1/2),x)

[Out]

x**15/(12*sqrt(x**6 + 2)) - x**9/(12*sqrt(x**6 + 2)) - x**3/(2*sqrt(x**6 + 2)) +
asinh(sqrt(2)*x**3/2)/2

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{\sqrt{x^{6} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/sqrt(x^6 + 2),x, algorithm="giac")

[Out]

integrate(x^14/sqrt(x^6 + 2), x)