3.1543 \(\int x^5 \sqrt{9+x^{12}} \, dx\)

Optimal. Leaf size=29 \[ \frac{3}{4} \sinh ^{-1}\left (\frac{x^6}{3}\right )+\frac{1}{12} \sqrt{x^{12}+9} x^6 \]

[Out]

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

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Rubi [A]  time = 0.0290317, antiderivative size = 29, 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 \[ \frac{3}{4} \sinh ^{-1}\left (\frac{x^6}{3}\right )+\frac{1}{12} \sqrt{x^{12}+9} x^6 \]

Antiderivative was successfully verified.

[In]  Int[x^5*Sqrt[9 + x^12],x]

[Out]

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

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Rubi in Sympy [A]  time = 2.79299, size = 22, normalized size = 0.76 \[ \frac{x^{6} \sqrt{x^{12} + 9}}{12} + \frac{3 \operatorname{asinh}{\left (\frac{x^{6}}{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(x**12+9)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.0131558, size = 29, normalized size = 1. \[ \frac{3}{4} \sinh ^{-1}\left (\frac{x^6}{3}\right )+\frac{1}{12} \sqrt{x^{12}+9} x^6 \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*Sqrt[9 + x^12],x]

[Out]

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

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Maple [A]  time = 0.046, size = 22, normalized size = 0.8 \[{\frac{3}{4}{\it Arcsinh} \left ({\frac{{x}^{6}}{3}} \right ) }+{\frac{{x}^{6}}{12}\sqrt{{x}^{12}+9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(x^12+9)^(1/2),x)

[Out]

3/4*arcsinh(1/3*x^6)+1/12*x^6*(x^12+9)^(1/2)

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Maxima [A]  time = 1.43614, size = 78, normalized size = 2.69 \[ \frac{3 \, \sqrt{x^{12} + 9}}{4 \, x^{6}{\left (\frac{x^{12} + 9}{x^{12}} - 1\right )}} + \frac{3}{8} \, \log \left (\frac{\sqrt{x^{12} + 9}}{x^{6}} + 1\right ) - \frac{3}{8} \, \log \left (\frac{\sqrt{x^{12} + 9}}{x^{6}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^12 + 9)*x^5,x, algorithm="maxima")

[Out]

3/4*sqrt(x^12 + 9)/(x^6*((x^12 + 9)/x^12 - 1)) + 3/8*log(sqrt(x^12 + 9)/x^6 + 1)
- 3/8*log(sqrt(x^12 + 9)/x^6 - 1)

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Fricas [A]  time = 0.23686, size = 120, normalized size = 4.14 \[ -\frac{2 \, x^{24} + 18 \, x^{12} + 9 \,{\left (2 \, x^{12} - 2 \, \sqrt{x^{12} + 9} x^{6} + 9\right )} \log \left (-x^{6} + \sqrt{x^{12} + 9}\right ) -{\left (2 \, x^{18} + 9 \, x^{6}\right )} \sqrt{x^{12} + 9}}{12 \,{\left (2 \, x^{12} - 2 \, \sqrt{x^{12} + 9} x^{6} + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^12 + 9)*x^5,x, algorithm="fricas")

[Out]

-1/12*(2*x^24 + 18*x^12 + 9*(2*x^12 - 2*sqrt(x^12 + 9)*x^6 + 9)*log(-x^6 + sqrt(
x^12 + 9)) - (2*x^18 + 9*x^6)*sqrt(x^12 + 9))/(2*x^12 - 2*sqrt(x^12 + 9)*x^6 + 9
)

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Sympy [A]  time = 5.5537, size = 37, normalized size = 1.28 \[ \frac{x^{18}}{12 \sqrt{x^{12} + 9}} + \frac{3 x^{6}}{4 \sqrt{x^{12} + 9}} + \frac{3 \operatorname{asinh}{\left (\frac{x^{6}}{3} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(x**12+9)**(1/2),x)

[Out]

x**18/(12*sqrt(x**12 + 9)) + 3*x**6/(4*sqrt(x**12 + 9)) + 3*asinh(x**6/3)/4

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GIAC/XCAS [A]  time = 0.226685, size = 39, normalized size = 1.34 \[ \frac{1}{12} \, \sqrt{x^{12} + 9} x^{6} - \frac{3}{4} \,{\rm ln}\left (-x^{6} + \sqrt{x^{12} + 9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x^12 + 9)*x^5,x, algorithm="giac")

[Out]

1/12*sqrt(x^12 + 9)*x^6 - 3/4*ln(-x^6 + sqrt(x^12 + 9))