3.146 \(\int \frac{x^2}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx\)

Optimal. Leaf size=248 \[ \frac{(-1)^{2/3} \tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{27\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac{(-1)^{2/3} \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{81 \sqrt [3]{2} \sqrt [6]{3} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{81\ 2^{5/6} \sqrt [6]{3} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]

[Out]

((-1)^(2/3)*ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3)
)]])/(27*2^(5/6)*3^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ((
-1)^(2/3)*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]
])/(81*2^(1/3)*3^(1/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3)]) - Ar
cTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(81*2^
(5/6)*3^(1/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)])

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Rubi [A]  time = 1.32345, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{(-1)^{2/3} \tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{27\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac{(-1)^{2/3} \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{81 \sqrt [3]{2} \sqrt [6]{3} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{81\ 2^{5/6} \sqrt [6]{3} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]

[Out]

((-1)^(2/3)*ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3)
)]])/(27*2^(5/6)*3^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ((
-1)^(2/3)*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]
])/(81*2^(1/3)*3^(1/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3)]) - Ar
cTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(81*2^
(5/6)*3^(1/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)])

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(x**6+18*x**4+324*x**3+108*x**2+216),x)

[Out]

Timed out

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Mathematica [C]  time = 0.019311, size = 59, normalized size = 0.24 \[ \frac{1}{6} \text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\&,\frac{\text{$\#$1} \log (x-\text{$\#$1})}{\text{$\#$1}^4+12 \text{$\#$1}^2+162 \text{$\#$1}+36}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]

[Out]

RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (Log[x - #1]*#1)/(36 + 16
2*#1 + 12*#1^2 + #1^4) & ]/6

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Maple [C]  time = 0.008, size = 56, normalized size = 0.2 \[{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{{{\it \_R}}^{2}\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(x^6+18*x^4+324*x^3+108*x^2+216),x)

[Out]

1/6*sum(_R^2/(_R^5+12*_R^3+162*_R^2+36*_R)*ln(x-_R),_R=RootOf(_Z^6+18*_Z^4+324*_
Z^3+108*_Z^2+216))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216),x, algorithm="maxima")

[Out]

integrate(x^2/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 0.485465, size = 48, normalized size = 0.19 \[ \operatorname{RootSum}{\left (732274264442769408 t^{6} - 2677850419968 t^{4} + 2834352 t^{2} - 1, \left ( t \mapsto t \log{\left (10170475895038464 t^{5} - 5231726283456 t^{4} - 31809932496 t^{3} + 19131876 t^{2} + 19683 t + x - \frac{27}{2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(x**6+18*x**4+324*x**3+108*x**2+216),x)

[Out]

RootSum(732274264442769408*_t**6 - 2677850419968*_t**4 + 2834352*_t**2 - 1, Lamb
da(_t, _t*log(10170475895038464*_t**5 - 5231726283456*_t**4 - 31809932496*_t**3
+ 19131876*_t**2 + 19683*_t + x - 27/2)))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216),x, algorithm="giac")

[Out]

integrate(x^2/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)