3.1413 \(\int \frac{x^{14}}{\left (2+x^6\right )^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ -\sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )-\frac{x^9}{3 \sqrt{x^6+2}}+\frac{1}{2} \sqrt{x^6+2} x^3 \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.055127, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )-\frac{x^9}{3 \sqrt{x^6+2}}+\frac{1}{2} \sqrt{x^6+2} x^3 \]

Antiderivative was successfully verified.

[In]  Int[x^14/(2 + x^6)^(3/2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.49788, size = 37, normalized size = 0.82 \[ - \frac{x^{9}}{3 \sqrt{x^{6} + 2}} + \frac{x^{3} \sqrt{x^{6} + 2}}{2} - \operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(x**6+2)**(3/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0386232, size = 43, normalized size = 0.96 \[ \frac{x^9+6 x^3-6 \sqrt{x^6+2} \sinh ^{-1}\left (\frac{x^3}{\sqrt{2}}\right )}{6 \sqrt{x^6+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/(2 + x^6)^(3/2),x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.034, size = 30, normalized size = 0.7 \[{\frac{{x}^{3} \left ({x}^{6}+6 \right ) }{6}{\frac{1}{\sqrt{{x}^{6}+2}}}}-{\it Arcsinh} \left ({\frac{{x}^{3}\sqrt{2}}{2}} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(x^6+2)^(3/2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.436, size = 99, normalized size = 2.2 \[ -\frac{\frac{3 \,{\left (x^{6} + 2\right )}}{x^{6}} - 2}{3 \,{\left (\frac{\sqrt{x^{6} + 2}}{x^{3}} - \frac{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}{x^{9}}\right )}} - \frac{1}{2} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} + 1\right ) + \frac{1}{2} \, \log \left (\frac{\sqrt{x^{6} + 2}}{x^{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(x^6 + 2)^(3/2),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.220444, size = 170, normalized size = 3.78 \[ -\frac{2 \, x^{18} + 7 \, x^{12} - 2 \, x^{6} - 6 \,{\left (2 \, x^{12} + 5 \, x^{6} -{\left (2 \, x^{9} + 3 \, x^{3}\right )} \sqrt{x^{6} + 2} + 2\right )} \log \left (-x^{3} + \sqrt{x^{6} + 2}\right ) -{\left (2 \, x^{15} + 5 \, x^{9} - 6 \, x^{3}\right )} \sqrt{x^{6} + 2} - 8}{6 \,{\left (2 \, x^{12} + 5 \, x^{6} -{\left (2 \, x^{9} + 3 \, x^{3}\right )} \sqrt{x^{6} + 2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(x^6 + 2)^(3/2),x, algorithm="fricas")

[Out]

-1/6*(2*x^18 + 7*x^12 - 2*x^6 - 6*(2*x^12 + 5*x^6 - (2*x^9 + 3*x^3)*sqrt(x^6 + 2
) + 2)*log(-x^3 + sqrt(x^6 + 2)) - (2*x^15 + 5*x^9 - 6*x^3)*sqrt(x^6 + 2) - 8)/(
2*x^12 + 5*x^6 - (2*x^9 + 3*x^3)*sqrt(x^6 + 2) + 2)

_______________________________________________________________________________________

Sympy [A]  time = 14.939, size = 36, normalized size = 0.8 \[ \frac{x^{9}}{6 \sqrt{x^{6} + 2}} + \frac{x^{3}}{\sqrt{x^{6} + 2}} - \operatorname{asinh}{\left (\frac{\sqrt{2} x^{3}}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(x**6+2)**(3/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/(x^6 + 2)^(3/2),x, algorithm="giac")

[Out]

integrate(x^14/(x^6 + 2)^(3/2), x)