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

Optimal. Leaf size=377 \[ \frac{\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\sqrt [3]{-\frac{1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{18\ 2^{2/3}}-\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{18\ 2^{2/3} \sqrt [3]{3}}+\frac{(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{9 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\frac{\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{27 \sqrt{3 \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac{\left (9-2^{2/3} \sqrt [3]{3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{27 \sqrt{6 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}} \]

[Out]

((-1)^(2/3)*(3*(-3)^(2/3) - 2^(2/3))*ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*
(4 - 3*(-3)^(2/3)*2^(1/3))]])/(9*3^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[2*(4 - 3*(-3)^(
2/3)*2^(1/3))]) + ((9 - (-2)^(2/3)*3^(1/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/
Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(27*Sqrt[3*(8 + (9*I)*2^(1/3)*3^(1/6) + 3*2
^(1/3)*3^(2/3))]) - ((9 - 2^(2/3)*3^(1/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)
*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]])/(27*Sqrt[6*(-4 + 3*2^(1/3)*3^(2/3))]) +
Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2]/(6*2^(2/3)*3^(1/3)*(1 + (-1)^(1/3))^2) + (
(-1/3)^(1/3)*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(18*2^(2/3)) - Log[6 + 3*2^(
2/3)*3^(1/3)*x + x^2]/(18*2^(2/3)*3^(1/3))

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Rubi [A]  time = 2.74215, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{6\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2}+\frac{\sqrt [3]{-\frac{1}{3}} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{18\ 2^{2/3}}-\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{18\ 2^{2/3} \sqrt [3]{3}}+\frac{(-1)^{2/3} \left (3 (-3)^{2/3}-2^{2/3}\right ) \tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{9 \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac{\left ((-2)^{2/3}-3\ 3^{2/3}\right ) \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{27 \sqrt [6]{3} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac{\left (2^{2/3}-3\ 3^{2/3}\right ) \tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{27 \sqrt [6]{3} \sqrt{2 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}} \]

Antiderivative was successfully verified.

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

[Out]

((-1)^(2/3)*(3*(-3)^(2/3) - 2^(2/3))*ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*
(4 - 3*(-3)^(2/3)*2^(1/3))]])/(9*3^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[2*(4 - 3*(-3)^(
2/3)*2^(1/3))]) - (((-2)^(2/3) - 3*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/
Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(27*3^(1/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6)
+ 3*2^(1/3)*3^(2/3)]) + ((2^(2/3) - 3*3^(2/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(
1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]])/(27*3^(1/6)*Sqrt[2*(-4 + 3*2^(1/3)*3
^(2/3))]) + Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2]/(6*2^(2/3)*3^(1/3)*(1 + (-1)^(
1/3))^2) + ((-1/3)^(1/3)*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(18*2^(2/3)) - L
og[6 + 3*2^(2/3)*3^(1/3)*x + x^2]/(18*2^(2/3)*3^(1/3))

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

[Out]

Timed out

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Mathematica [C]  time = 0.0203644, size = 61, normalized size = 0.16 \[ \frac{1}{6} \text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\&,\frac{\text{$\#$1}^3 \log (x-\text{$\#$1})}{\text{$\#$1}^4+12 \text{$\#$1}^2+162 \text{$\#$1}+36}\&\right ] \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(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]*#1^3)/(36 +
162*#1 + 12*#1^2 + #1^4) & ]/6

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Maple [C]  time = 0.007, size = 56, normalized size = 0.2 \[{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{{{\it \_R}}^{4}\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*sum(_R^4/(_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. \[ \int \frac{x^{4}}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(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^4/(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 = 0.710731, size = 65, normalized size = 0.17 \[ \operatorname{RootSum}{\left (15695178850368 t^{6} - 2066242608 t^{4} + 1845163152 t^{3} - 1180980 t^{2} - 1944 t - 1, \left ( t \mapsto t \log{\left (\frac{614714526178551746208 t^{5}}{57121295165} - \frac{1270857362386176 t^{4}}{57121295165} - \frac{80483053187684376 t^{3}}{57121295165} + \frac{72431318325103884 t^{2}}{57121295165} - \frac{45358602689088 t}{57121295165} + x - \frac{44532180783}{57121295165} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(15695178850368*_t**6 - 2066242608*_t**4 + 1845163152*_t**3 - 1180980*_t*
*2 - 1944*_t - 1, Lambda(_t, _t*log(614714526178551746208*_t**5/57121295165 - 12
70857362386176*_t**4/57121295165 - 80483053187684376*_t**3/57121295165 + 7243131
8325103884*_t**2/57121295165 - 45358602689088*_t/57121295165 + x - 44532180783/5
7121295165)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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