3.1414 \(\int \frac{x^8}{\left (2+x^6\right )^{3/2}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0352522, 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 \[ \frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )-\frac{x^3}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(2 + x^6)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.45744, size = 26, normalized size = 0.84 \[ - \frac{x^{3}}{3 \sqrt{x^{6} + 2}} + \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(x**6+2)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0236983, size = 31, normalized size = 1. \[ \frac{1}{3} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )-\frac{x^3}{3 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(2 + x^6)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.019, size = 25, normalized size = 0.8 \[{\frac{1}{3}{\it Arcsinh} \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ) }-{\frac{{x}^{3}}{3}{\frac{1}{\sqrt{{x}^{6}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(x^6+2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.44025, size = 61, normalized size = 1.97 \[ -\frac{x^{3}}{3 \, \sqrt{x^{6} + 2}} + \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} + 1\right ) - \frac{1}{6} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(x^6 + 2)^(3/2),x, algorithm="maxima")

[Out]

-1/3*x^3/sqrt(x^6 + 2) + 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 = 0.220378, size = 74, normalized size = 2.39 \[ -\frac{{\left (x^{6} - \sqrt{x^{6} + 2} x^{3} + 2\right )} \log \left (-x^{3} + \sqrt{x^{6} + 2}\right ) + 2}{3 \,{\left (x^{6} - \sqrt{x^{6} + 2} x^{3} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(x^6 + 2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.63531, size = 26, normalized size = 0.84 \[ - \frac{x^{3}}{3 \sqrt{x^{6} + 2}} + \frac{\operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(x**6+2)**(3/2),x)

[Out]

-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. \[ \int \frac{x^{8}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(x^6 + 2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^8/(x^6 + 2)^(3/2), x)