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

Optimal. Leaf size=986 \[ \text{result too large to display} \]

[Out]

-(27*((-2)^(2/3) + 2*(-1)^(1/3)*3^(2/3)) - 6^(1/3)*(9 + (-3)^(1/3)*2^(2/3))*x)/(
104976*2^(2/3)*(1 + (-1)^(1/3))^4*(4 - 3*(-3)^(2/3)*2^(1/3))*(6 - 3*(-3)^(1/3)*2
^(2/3)*x + x^2)) - (27*2^(2/3)*(1 + (-2)^(1/3)*3^(2/3)) - (-1)^(1/3)*3^(2/3)*(2
+ 3*(-2)^(1/3)*3^(2/3))*x)/(472392*2^(2/3)*(8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3
)*3^(2/3))*(6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2)) + (9*(6 - 2^(2/3)*3^(1/3)) - (2 -
3*2^(1/3)*3^(2/3))*x)/(314928*2^(2/3)*3^(1/3)*(4 - 3*2^(1/3)*3^(2/3))*(6 + 3*2^
(2/3)*3^(1/3)*x + x^2)) - ((1 + I*Sqrt[3] + 3*2^(1/3)*3^(2/3))*ArcTan[(3*(-3)^(1
/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(8748*2^(2/3)*3^(5/6)*(1
+ (-1)^(1/3))^4*(8 - (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) + ((3*(-
3)^(2/3) + (-1)^(1/3)*2^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3
*(-2)^(1/3)*3^(2/3))]])/(17496*6^(5/6)*(1 - (-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(4
+ 3*(-2)^(1/3)*3^(2/3))^(3/2)) + ((I + Sqrt[3])*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2
*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(34992*2^(1/6)*3^(1/3)*(1 + (-1)^(1/3))
^5*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) + ((I/17496)*ArcTan[(2^(1/6)*(3*(-3)^(1/3) -
2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(2^(1/6)*3^(1/3)*(1 + (-1)^(1/3
))^5*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((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))]])/(17496*6^(5/6)*(1 - (
-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) - ArcTanh[(2^(1/
6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(157464*2^(1/6)*3^
(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + ((I + Sqrt[3])*Log[6 - 3*(-3)^(1/3)*2^(2/3
)*x + x^2])/(419904*2^(1/3)*3^(1/6)*(1 + (-1)^(1/3))^5) - ((I/209952)*Log[6 + 3*
(-2)^(2/3)*3^(1/3)*x + x^2])/(2^(1/3)*3^(1/6)*(1 + (-1)^(1/3))^5) + Log[6 + 3*2^
(2/3)*3^(1/3)*x + x^2]/(1889568*2^(1/3)*3^(2/3))

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Rubi [A]  time = 7.46294, antiderivative size = 986, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269 \[ -\frac{27 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right )-\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{104976\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac{\left (1+i \sqrt{3}+3 \sqrt [3]{2} 3^{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 )}{8748\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac{\left (i+\sqrt{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 )}{34992 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac{\left (3 (-3)^{2/3}+\sqrt [3]{-1} 2^{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 )}{17496\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}+\frac{i \tan ^{-1}\left (\frac{\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt{3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{17496 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt{4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt{3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{157464 \sqrt [6]{2} 3^{5/6} \sqrt{-4+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 (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{17496\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac{\left (i+\sqrt{3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{419904 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}-\frac{i \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{209952 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac{\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{1889568 \sqrt [3]{2} 3^{2/3}}-\frac{27\ 2^{2/3} \left (1+\sqrt [3]{-2} 3^{2/3}\right )-\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{472392\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac{9 \left (6-2^{2/3} \sqrt [3]{3}\right )-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{314928\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(27*((-2)^(2/3) + 2*(-1)^(1/3)*3^(2/3)) - 6^(1/3)*(9 + (-3)^(1/3)*2^(2/3))*x)/(
104976*2^(2/3)*(1 + (-1)^(1/3))^4*(4 - 3*(-3)^(2/3)*2^(1/3))*(6 - 3*(-3)^(1/3)*2
^(2/3)*x + x^2)) - (27*2^(2/3)*(1 + (-2)^(1/3)*3^(2/3)) - (-1)^(1/3)*3^(2/3)*(2
+ 3*(-2)^(1/3)*3^(2/3))*x)/(472392*2^(2/3)*(8 + (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3
)*3^(2/3))*(6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2)) + (9*(6 - 2^(2/3)*3^(1/3)) - (2 -
3*2^(1/3)*3^(2/3))*x)/(314928*2^(2/3)*3^(1/3)*(4 - 3*2^(1/3)*3^(2/3))*(6 + 3*2^
(2/3)*3^(1/3)*x + x^2)) - ((1 + I*Sqrt[3] + 3*2^(1/3)*3^(2/3))*ArcTan[(3*(-3)^(1
/3)*2^(2/3) - 2*x)/Sqrt[6*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(8748*2^(2/3)*3^(5/6)*(1
+ (-1)^(1/3))^4*(8 - (9*I)*2^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))^(3/2)) + ((3*(-
3)^(2/3) + (-1)^(1/3)*2^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3
*(-2)^(1/3)*3^(2/3))]])/(17496*6^(5/6)*(1 - (-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(4
+ 3*(-2)^(1/3)*3^(2/3))^(3/2)) + ((I + Sqrt[3])*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2
*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(34992*2^(1/6)*3^(1/3)*(1 + (-1)^(1/3))
^5*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) + ((I/17496)*ArcTan[(2^(1/6)*(3*(-3)^(1/3) -
2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(2^(1/6)*3^(1/3)*(1 + (-1)^(1/3
))^5*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((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))]])/(17496*6^(5/6)*(1 - (
-1)^(1/3))^2*(1 + (-1)^(1/3))^4*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) - ArcTanh[(2^(1/
6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(157464*2^(1/6)*3^
(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + ((I + Sqrt[3])*Log[6 - 3*(-3)^(1/3)*2^(2/3
)*x + x^2])/(419904*2^(1/3)*3^(1/6)*(1 + (-1)^(1/3))^5) - ((I/209952)*Log[6 + 3*
(-2)^(2/3)*3^(1/3)*x + x^2])/(2^(1/3)*3^(1/6)*(1 + (-1)^(1/3))^5) + Log[6 + 3*2^
(2/3)*3^(1/3)*x + x^2]/(1889568*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)**2,x)

[Out]

Timed out

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Mathematica [C]  time = 0.0543622, size = 167, normalized size = 0.17 \[ \frac{-9 x^5+8 x^4-216 x^3-2724 x^2+324 x-7884}{7383312 \left (x^6+18 x^4+324 x^3+108 x^2+216\right )}-\frac{\text{RootSum}\left [\text{$\#$1}^6+18 \text{$\#$1}^4+324 \text{$\#$1}^3+108 \text{$\#$1}^2+216\&,\frac{9 \text{$\#$1}^4 \log (x-\text{$\#$1})-16 \text{$\#$1}^3 \log (x-\text{$\#$1})+324 \text{$\#$1}^2 \log (x-\text{$\#$1})+2436 \text{$\#$1} \log (x-\text{$\#$1})+324 \log (x-\text{$\#$1})}{\text{$\#$1}^5+12 \text{$\#$1}^3+162 \text{$\#$1}^2+36 \text{$\#$1}}\&\right ]}{44299872} \]

Antiderivative was successfully verified.

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

[Out]

(-7884 + 324*x - 2724*x^2 - 216*x^3 + 8*x^4 - 9*x^5)/(7383312*(216 + 108*x^2 + 3
24*x^3 + 18*x^4 + x^6)) - RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & ,
(324*Log[x - #1] + 2436*Log[x - #1]*#1 + 324*Log[x - #1]*#1^2 - 16*Log[x - #1]*
#1^3 + 9*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/44299872

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Maple [C]  time = 0.015, size = 122, normalized size = 0.1 \[{\frac{1}{{x}^{6}+18\,{x}^{4}+324\,{x}^{3}+108\,{x}^{2}+216} \left ( -{\frac{{x}^{5}}{820368}}+{\frac{{x}^{4}}{922914}}-{\frac{{x}^{3}}{34182}}-{\frac{227\,{x}^{2}}{615276}}+{\frac{x}{22788}}-{\frac{73}{68364}} \right ) }+{\frac{1}{44299872}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}+18\,{{\it \_Z}}^{4}+324\,{{\it \_Z}}^{3}+108\,{{\it \_Z}}^{2}+216 \right ) }{\frac{ \left ( -9\,{{\it \_R}}^{4}+16\,{{\it \_R}}^{3}-324\,{{\it \_R}}^{2}-2436\,{\it \_R}-324 \right ) \ln \left ( x-{\it \_R} \right ) }{{{\it \_R}}^{5}+12\,{{\it \_R}}^{3}+162\,{{\it \_R}}^{2}+36\,{\it \_R}}}} \]

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)^2,x)

[Out]

(-1/820368*x^5+1/922914*x^4-1/34182*x^3-227/615276*x^2+1/22788*x-73/68364)/(x^6+
18*x^4+324*x^3+108*x^2+216)+1/44299872*sum((-9*_R^4+16*_R^3-324*_R^2-2436*_R-324
)/(_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. \[ -\frac{9 \, x^{5} - 8 \, x^{4} + 216 \, x^{3} + 2724 \, x^{2} - 324 \, x + 7884}{7383312 \,{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}} - \frac{1}{7383312} \, \int \frac{9 \, x^{4} - 16 \, x^{3} + 324 \, x^{2} + 2436 \, x + 324}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216}\,{d x} \]

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)^2,x, algorithm="maxima")

[Out]

-1/7383312*(9*x^5 - 8*x^4 + 216*x^3 + 2724*x^2 - 324*x + 7884)/(x^6 + 18*x^4 + 3
24*x^3 + 108*x^2 + 216) - 1/7383312*integrate((9*x^4 - 16*x^3 + 324*x^2 + 2436*x
+ 324)/(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)^2,x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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Sympy [A]  time = 1.19159, size = 112, normalized size = 0.11 \[ \operatorname{RootSum}{\left (8658597397620778437929792538933565560629231616 t^{6} + 109068095871770168248838645612544 t^{4} - 492655707593366915713499136 t^{3} + 40378331745144603648 t^{2} - 695635011360 t + 4513, \left ( t \mapsto t \log{\left (\frac{101442531561804181113161287039859349851881619653631712165888 t^{5}}{356900697070792948475845} - \frac{149796550082359335112709434971975088967050210050048 t^{4}}{356900697070792948475845} + \frac{1222409754458272818505898777768670783617236992 t^{3}}{356900697070792948475845} - \frac{5775055524251595723022901938558261453824 t^{2}}{356900697070792948475845} + \frac{96165242200260265765603930470432 t}{71380139414158589695169} + x - \frac{17059152341129698120545584}{1070702091212378845427535} \right )} \right )\right )} - \frac{9 x^{5} - 8 x^{4} + 216 x^{3} + 2724 x^{2} - 324 x + 7884}{7383312 x^{6} + 132899616 x^{4} + 2392193088 x^{3} + 797397696 x^{2} + 1594795392} \]

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)**2,x)

[Out]

RootSum(8658597397620778437929792538933565560629231616*_t**6 + 10906809587177016
8248838645612544*_t**4 - 492655707593366915713499136*_t**3 + 4037833174514460364
8*_t**2 - 695635011360*_t + 4513, Lambda(_t, _t*log(1014425315618041811131612870
39859349851881619653631712165888*_t**5/356900697070792948475845 - 14979655008235
9335112709434971975088967050210050048*_t**4/356900697070792948475845 + 122240975
4458272818505898777768670783617236992*_t**3/356900697070792948475845 - 577505552
4251595723022901938558261453824*_t**2/356900697070792948475845 + 961652422002602
65765603930470432*_t/71380139414158589695169 + x - 17059152341129698120545584/10
70702091212378845427535))) - (9*x**5 - 8*x**4 + 216*x**3 + 2724*x**2 - 324*x + 7
884)/(7383312*x**6 + 132899616*x**4 + 2392193088*x**3 + 797397696*x**2 + 1594795
392)

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

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)^2,x, algorithm="giac")

[Out]

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