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

Optimal. Leaf size=361 \[ \frac{(-1)^{2/3} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{(-1)^{2/3} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{36 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac{\sqrt [3]{-1} \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{54\ 2^{2/3} 3^{5/6} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{108 \sqrt [6]{2} 3^{5/6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]

[Out]

-ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]]/(36*2^(
1/6)*3^(5/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ((-1)^(1/3)*Ar
cTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(54*2^(2/
3)*3^(5/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*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))]]/(108*2^(1/6)*3^(5/
6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + ((-1)^(2/3)*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x +
x^2])/(216*2^(1/3)*3^(2/3)*(1 + (-1)^(1/3))^2) - ((-1)^(2/3)*Log[6 + 3*(-2)^(2/3
)*3^(1/3)*x + x^2])/(648*2^(1/3)*3^(2/3)) - Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2]/(
648*2^(1/3)*3^(2/3))

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Rubi [A]  time = 2.0054, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{(-1)^{2/3} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac{(-1)^{2/3} \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{648 \sqrt [3]{2} 3^{2/3}}-\frac{\tan ^{-1}\left (\frac{3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt{6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{36 \sqrt [6]{2} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac{\sqrt [3]{-1} \tan ^{-1}\left (\frac{2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt{6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{54\ 2^{2/3} 3^{5/6} \sqrt{8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{108 \sqrt [6]{2} 3^{5/6} \sqrt{3 \sqrt [3]{2} 3^{2/3}-4}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(3*(-3)^(1/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]]/(36*2^(
1/6)*3^(5/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ((-1)^(1/3)*Ar
cTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(54*2^(2/
3)*3^(5/6)*Sqrt[8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*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))]]/(108*2^(1/6)*3^(5/
6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + ((-1)^(2/3)*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x +
x^2])/(216*2^(1/3)*3^(2/3)*(1 + (-1)^(1/3))^2) - ((-1)^(2/3)*Log[6 + 3*(-2)^(2/3
)*3^(1/3)*x + x^2])/(648*2^(1/3)*3^(2/3)) - Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2]/(
648*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/(x**6+18*x**4+324*x**3+108*x**2+216),x)

[Out]

Timed out

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Mathematica [C]  time = 0.0186764, size = 57, normalized size = 0.16 \[ \frac{1}{6} \text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\&,\frac{\log (x-\text{$\#$1})}{\text{$\#$1}^4+12 \text{$\#$1}^2+162 \text{$\#$1}+36}\&\right ] \]

Antiderivative was successfully verified.

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

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Maple [C]  time = 0.007, size = 54, normalized size = 0.2 \[{\frac{1}{6}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{{\it \_R}\,\ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(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/(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 = 0.691747, size = 61, normalized size = 0.17 \[ \operatorname{RootSum}{\left (158171241119638192128 t^{6} - 96402615118848 t^{4} + 287743415040 t^{3} - 51018336 t^{2} - 1, \left ( t \mapsto t \log{\left (\frac{65418399445721140961280 t^{5}}{415817} + \frac{2480926457425102848 t^{4}}{415817} - \frac{39451802929737984 t^{3}}{415817} + \frac{118071997444800 t^{2}}{415817} - \frac{16745884920 t}{415817} + x - \frac{268790}{415817} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

RootSum(158171241119638192128*_t**6 - 96402615118848*_t**4 + 287743415040*_t**3
- 51018336*_t**2 - 1, Lambda(_t, _t*log(65418399445721140961280*_t**5/415817 + 2
480926457425102848*_t**4/415817 - 39451802929737984*_t**3/415817 + 1180719974448
00*_t**2/415817 - 16745884920*_t/415817 + x - 268790/415817)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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