3.1389 \(\int \frac{x^8}{\sqrt{2+x^6}} \, dx\)

Optimal. Leaf size=31 \[ \frac{1}{6} x^3 \sqrt{x^6+2}-\frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0357034, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{1}{6} x^3 \sqrt{x^6+2}-\frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[x^8/Sqrt[2 + x^6],x]

[Out]

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

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Rubi in Sympy [A]  time = 4.68166, size = 26, normalized size = 0.84 \[ \frac{x^{3} \sqrt{x^{6} + 2}}{6} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(x**6+2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.01697, size = 31, normalized size = 1. \[ \frac{1}{6} x^3 \sqrt{x^6+2}-\frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/Sqrt[2 + x^6],x]

[Out]

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

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Maple [A]  time = 0.034, size = 25, normalized size = 0.8 \[ -{\frac{1}{3}{\it Arcsinh} \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ) }+{\frac{{x}^{3}}{6}\sqrt{{x}^{6}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(x^6+2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.43799, size = 78, normalized size = 2.52 \[ \frac{\sqrt{x^{6} + 2}}{3 \, x^{3}{\left (\frac{x^{6} + 2}{x^{6}} - 1\right )}} - \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} + 1\right ) + \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(x^6 + 2),x, algorithm="maxima")

[Out]

1/3*sqrt(x^6 + 2)/(x^3*((x^6 + 2)/x^6 - 1)) - 1/6*log(sqrt(x^6 + 2)/x^3 + 1) + 1
/6*log(sqrt(x^6 + 2)/x^3 - 1)

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Fricas [A]  time = 0.223254, size = 107, normalized size = 3.45 \[ -\frac{x^{12} + 2 \, x^{6} - 2 \,{\left (x^{6} - \sqrt{x^{6} + 2} x^{3} + 1\right )} \log \left (-x^{3} + \sqrt{x^{6} + 2}\right ) -{\left (x^{9} + x^{3}\right )} \sqrt{x^{6} + 2}}{6 \,{\left (x^{6} - \sqrt{x^{6} + 2} x^{3} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(x^6 + 2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.70722, size = 39, normalized size = 1.26 \[ \frac{x^{9}}{6 \sqrt{x^{6} + 2}} + \frac{x^{3}}{3 \sqrt{x^{6} + 2}} - \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(x**6+2)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{x^{6} + 2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(x^6 + 2),x, algorithm="giac")

[Out]

integrate(x^8/sqrt(x^6 + 2), x)