Integrand size = 26, antiderivative size = 986 \[ \int \frac {x^2}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx =\text {Too large to display} \] Output:
-1/209952*(27*(-2)^(2/3)+54*(-1)^(1/3)*3^(2/3)-6^(1/3)*(9+(-3)^(1/3)*2^(2/ 3))*x)*2^(1/3)/(1+(-1)^(1/3))^4/(4-3*(-3)^(2/3)*2^(1/3))/(6-3*(-3)^(1/3)*2 ^(2/3)*x+x^2)-1/944784*(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)*2^(1/3)/(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)+1/1889568*(54-9*2^(2/3)*3^(1/3)-(2-3 *2^(1/3)*3^(2/3))*x)*2^(1/3)*3^(2/3)/(4-3*2^(1/3)*3^(2/3))/(6+3*2^(2/3)*3^ (1/3)*x+x^2)-1/52488*(1+I*3^(1/2)+3*2^(1/3)*3^(2/3))*arctan((3*(-3)^(1/3)* 2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2))*2^(1/3)*3^(1/6)/(1+(-1)^(1/ 3))^4/(8-9*I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(3/2)+1/104976*(3*(-3)^(2/ 3)+(-1)^(1/3)*2^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3) *3^(2/3))^(1/2))*6^(1/6)/(1-(-1)^(1/3))^2/(1+(-1)^(1/3))^4/(4+3*(-2)^(1/3) *3^(2/3))^(3/2)+1/209952*(3^(1/2)+I)*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24 +18*(-2)^(1/3)*3^(2/3))^(1/2))*2^(5/6)*3^(2/3)/(1+(-1)^(1/3))^5/(4+3*(-2)^ (1/3)*3^(2/3))^(1/2)+1/104976*I*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/(1 2-9*(-3)^(2/3)*2^(1/3))^(1/2))*2^(5/6)*3^(2/3)/(1+(-1)^(1/3))^5/(4-3*(-3)^ (2/3)*2^(1/3))^(1/2)-1/104976*(2^(2/3)-3*3^(2/3))*arctanh(2^(1/6)*(3*3^(1/ 3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*6^(1/6)/(1-(-1)^(1/3))^2/(1+( -1)^(1/3))^4/(-4+3*2^(1/3)*3^(2/3))^(3/2)-1/944784*arctanh(2^(1/6)*(3*3^(1 /3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*2^(5/6)*3^(1/6)/(-4+3*2^(1/3 )*3^(2/3))^(1/2)+1/2519424*(3^(1/2)+I)*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)...
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.17 \[ \int \frac {x^2}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {-7884+324 x-2724 x^2-216 x^3+8 x^4-9 x^5}{7383312 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {324 \log (x-\text {$\#$1})+2436 \log (x-\text {$\#$1}) \text {$\#$1}+324 \log (x-\text {$\#$1}) \text {$\#$1}^2-16 \log (x-\text {$\#$1}) \text {$\#$1}^3+9 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{44299872} \] Input:
Integrate[x^2/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]
Output:
(-7884 + 324*x - 2724*x^2 - 216*x^3 + 8*x^4 - 9*x^5)/(7383312*(216 + 108*x ^2 + 324*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
Time = 4.00 (sec) , antiderivative size = 892, normalized size of antiderivative = 0.90, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2466, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2}{\left (x^6+18 x^4+324 x^3+108 x^2+216\right )^2} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1586874322944 \int \left (\frac {3 x+\sqrt [3]{-6} \left (9 \sqrt [3]{-2}+\sqrt [3]{3}\right )}{499751156776108032 \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )^2}-\frac {i \left (18 \sqrt [3]{3}-\sqrt [3]{2} \left (1-i \sqrt {3}\right ) x\right )}{333167437850738688\ 2^{2/3} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {9 \left (1+\sqrt [3]{-1}\right )-i \sqrt [3]{2} \sqrt [6]{3} x}{166583718925369344\ 2^{2/3} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {\sqrt [3]{2} x+9 \sqrt [3]{3}}{1499253470328324096\ 6^{2/3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac {(-6)^{2/3} x+2 \left (\sqrt [3]{-3}+9 \sqrt [3]{2}\right )}{18509302102818816\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )^2}-\frac {\sqrt [3]{2} \left (1-3 \sqrt [3]{2} 3^{2/3}\right )-\sqrt [3]{3} x}{166583718925369344 \sqrt [3]{3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1586874322944 \left (-\frac {9 \left ((-6)^{2/3}+6 \sqrt [3]{-3}\right )-\left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right ) x}{55527906308456448\ 6^{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 (i+\sqrt {3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{55527906308456448 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {\sqrt [3]{-1} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{499751156776108032 \sqrt [6]{2} 3^{5/6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (9+\sqrt [3]{-3} 2^{2/3}\right ) \arctan \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 )}{83291859462684672\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}+\frac {i \arctan \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 )}{27763953154228224 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\text {arctanh}\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 )}{249875578388054016 \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 ) \text {arctanh}\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 )}{249875578388054016\ 6^{5/6} \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 )}{666334875701477376 \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 )}{333167437850738688 \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 )}{2998506940656648192 \sqrt [3]{2} 3^{2/3}}-\frac {(-6)^{2/3} \left ((-2)^{2/3}-3\ 3^{2/3}\right ) x+54 \left (1+\sqrt [3]{-2} 3^{2/3}\right )}{2998506940656648192 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {18 \left (3 \sqrt [3]{2}-\sqrt [3]{3}\right )-\left (2 \sqrt [3]{2}-3\ 6^{2/3}\right ) x}{999502313552216064 \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 )}\right )\) |
Input:
Int[x^2/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]
Output:
1586874322944*(-1/55527906308456448*(9*((-6)^(2/3) + 6*(-3)^(1/3)) - (2*(- 3)^(1/3) + 9*2^(1/3))*x)/(6^(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)) - (54*(1 + (-2)^(1/3)*3^(2/3)) + (-6)^(2/3)*((-2)^(2/3) - 3*3^(2/3))*x)/(2998506940656648192*(4 + 3*(-2)^( 1/3)*3^(2/3))*(6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2)) + (18*(3*2^(1/3) - 3^(1/ 3)) - (2*2^(1/3) - 3*6^(2/3))*x)/(999502313552216064*3^(1/3)*(4 - 3*2^(1/3 )*3^(2/3))*(6 + 3*2^(2/3)*3^(1/3)*x + x^2)) + ((-1)^(1/3)*(2 + 3*(-2)^(1/3 )*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^ (2/3))]])/(499751156776108032*2^(1/6)*3^(5/6)*(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))]])/(55527906308456448*2^(1/6)*3^(1/3)*(1 + (-1)^(1/3))^ 5*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) + ((I/27763953154228224)*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)]) - ((9 + (-3)^( 1/3)*2^(2/3))*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(- 3)^(2/3)*2^(1/3))]])/(83291859462684672*2^(5/6)*3^(1/6)*(1 + (-1)^(1/3))^4 *(4 - 3*(-3)^(2/3)*2^(1/3))^(3/2)) - ((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))]])/(2498755783 88054016*6^(5/6)*(-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))]]/(24987557838805401...
Int[(u_.)*(Q6_)^(p_), x_Symbol] :> With[{a = Coeff[Q6, x, 0], b = Coeff[Q6, x, 2], c = Coeff[Q6, x, 3], d = Coeff[Q6, x, 4], e = Coeff[Q6, x, 6]}, Sim p[1/(3^(3*p)*a^(2*p)) Int[ExpandIntegrand[u*(3*a + 3*Rt[a, 3]^2*Rt[c, 3]* x + b*x^2)^p*(3*a - 3*(-1)^(1/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p*(3*a + 3* (-1)^(2/3)*Rt[a, 3]^2*Rt[c, 3]*x + b*x^2)^p, x], x], x] /; EqQ[b^2 - 3*a*d, 0] && EqQ[b^3 - 27*a^2*e, 0]] /; ILtQ[p, 0] && PolyQ[Q6, x, 6] && EqQ[Coef f[Q6, x, 1], 0] && EqQ[Coeff[Q6, x, 5], 0] && RationalFunctionQ[u, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.05 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.12
| method | result | size |
| default | \(\frac {-\frac {1}{820368} x^{5}+\frac {1}{922914} x^{4}-\frac {1}{34182} x^{3}-\frac {227}{615276} x^{2}+\frac {1}{22788} x -\frac {73}{68364}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-9 \textit {\_R}^{4}+16 \textit {\_R}^{3}-324 \textit {\_R}^{2}-2436 \textit {\_R} -324\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{44299872}\) | \(122\) |
| risch | \(\frac {-\frac {1}{820368} x^{5}+\frac {1}{922914} x^{4}-\frac {1}{34182} x^{3}-\frac {227}{615276} x^{2}+\frac {1}{22788} x -\frac {73}{68364}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-9 \textit {\_R}^{4}+16 \textit {\_R}^{3}-324 \textit {\_R}^{2}-2436 \textit {\_R} -324\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{44299872}\) | \(122\) |
Input:
int(x^2/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x,method=_RETURNVERBOSE)
Output:
(-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))
Timed out. \[ \int \frac {x^2}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^2/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="fricas")
Output:
Timed out
Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.11 \[ \int \frac {x^2}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\operatorname {RootSum} {\left (8658597397620778437929792538933565560629231616 t^{6} + 109068095871770168248838645612544 t^{4} - 492655707593366915713499136 t^{3} + 40378331745144603648 t^{2} - 695635011360 t + 4513, \left ( t \mapsto t \log {\left (\frac {101442531561804181113161287039859349851881619653631712165888 t^{5}}{356900697070792948475845} - \frac {149796550082359335112709434971975088967050210050048 t^{4}}{356900697070792948475845} + \frac {1222409754458272818505898777768670783617236992 t^{3}}{356900697070792948475845} - \frac {5775055524251595723022901938558261453824 t^{2}}{356900697070792948475845} + \frac {96165242200260265765603930470432 t}{71380139414158589695169} + x - \frac {17059152341129698120545584}{1070702091212378845427535} \right )} \right )\right )} + \frac {- 9 x^{5} + 8 x^{4} - 216 x^{3} - 2724 x^{2} + 324 x - 7884}{7383312 x^{6} + 132899616 x^{4} + 2392193088 x^{3} + 797397696 x^{2} + 1594795392} \] Input:
integrate(x**2/(x**6+18*x**4+324*x**3+108*x**2+216)**2,x)
Output:
RootSum(8658597397620778437929792538933565560629231616*_t**6 + 10906809587 1770168248838645612544*_t**4 - 492655707593366915713499136*_t**3 + 4037833 1745144603648*_t**2 - 695635011360*_t + 4513, Lambda(_t, _t*log(1014425315 61804181113161287039859349851881619653631712165888*_t**5/35690069707079294 8475845 - 149796550082359335112709434971975088967050210050048*_t**4/356900 697070792948475845 + 1222409754458272818505898777768670783617236992*_t**3/ 356900697070792948475845 - 5775055524251595723022901938558261453824*_t**2/ 356900697070792948475845 + 96165242200260265765603930470432*_t/71380139414 158589695169 + x - 17059152341129698120545584/1070702091212378845427535))) + (-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)
\[ \int \frac {x^2}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{2}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \] Input:
integrate(x^2/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="maxima")
Output:
-1/7383312*(9*x^5 - 8*x^4 + 216*x^3 + 2724*x^2 - 324*x + 7884)/(x^6 + 18*x ^4 + 324*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)
\[ \int \frac {x^2}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{2}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \] Input:
integrate(x^2/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="giac")
Output:
integrate(x^2/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)^2, x)
Time = 22.58 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.39 \[ \int \frac {x^2}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Too large to display} \] Input:
int(x^2/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216)^2,x)
Output:
symsum(log((4897*x)/18772949180387057928192 - (8147*root(z^6 + (163*z^4)/1 2940093659208032 - (8113597*z^3)/142599321974220092022546432 + (5171*z^2)/ 1108852327671535435567321055232 - (505*z)/62857556252809436097422151350778 59328 + 4513/8658597397620778437929792538933565560629231616, z, k))/110389 4245809216 - (1197643*root(z^6 + (163*z^4)/12940093659208032 - (8113597*z^ 3)/142599321974220092022546432 + (5171*z^2)/110885232767153543556732105523 2 - (505*z)/6285755625280943609742215135077859328 + 4513/86585973976207784 37929792538933565560629231616, z, k)*x)/29805144636848832 + (452809*root(z ^6 + (163*z^4)/12940093659208032 - (8113597*z^3)/1425993219742200920225464 32 + (5171*z^2)/1108852327671535435567321055232 - (505*z)/6285755625280943 609742215135077859328 + 4513/865859739762077843792979253893356556062923161 6, z, k)^2*x)/194734854 - (1241776944*root(z^6 + (163*z^4)/129400936592080 32 - (8113597*z^3)/142599321974220092022546432 + (5171*z^2)/11088523276715 35435567321055232 - (505*z)/6285755625280943609742215135077859328 + 4513/8 658597397620778437929792538933565560629231616, z, k)^3*x)/44521 + (4524079 28832*root(z^6 + (163*z^4)/12940093659208032 - (8113597*z^3)/1425993219742 20092022546432 + (5171*z^2)/1108852327671535435567321055232 - (505*z)/6285 755625280943609742215135077859328 + 4513/865859739762077843792979253893356 5560629231616, z, k)^4*x)/211 - 6940988288557056*root(z^6 + (163*z^4)/1294 0093659208032 - (8113597*z^3)/142599321974220092022546432 + (5171*z^2)/...
\[ \int \frac {x^2}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {too large to display} \] Input:
int(x^2/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x)
Output:
(14580*int(x**8/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11664*x**7 + 109 296*x**6 + 69984*x**5 + 19440*x**4 + 139968*x**3 + 46656*x**2 + 46656),x)* x**6 + 262440*int(x**8/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11664*x** 7 + 109296*x**6 + 69984*x**5 + 19440*x**4 + 139968*x**3 + 46656*x**2 + 466 56),x)*x**4 + 4723920*int(x**8/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 1 1664*x**7 + 109296*x**6 + 69984*x**5 + 19440*x**4 + 139968*x**3 + 46656*x* *2 + 46656),x)*x**3 + 1574640*int(x**8/(x**12 + 36*x**10 + 648*x**9 + 540* x**8 + 11664*x**7 + 109296*x**6 + 69984*x**5 + 19440*x**4 + 139968*x**3 + 46656*x**2 + 46656),x)*x**2 + 3149280*int(x**8/(x**12 + 36*x**10 + 648*x** 9 + 540*x**8 + 11664*x**7 + 109296*x**6 + 69984*x**5 + 19440*x**4 + 139968 *x**3 + 46656*x**2 + 46656),x) + 184680*int(x**6/(x**12 + 36*x**10 + 648*x **9 + 540*x**8 + 11664*x**7 + 109296*x**6 + 69984*x**5 + 19440*x**4 + 1399 68*x**3 + 46656*x**2 + 46656),x)*x**6 + 3324240*int(x**6/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11664*x**7 + 109296*x**6 + 69984*x**5 + 19440*x** 4 + 139968*x**3 + 46656*x**2 + 46656),x)*x**4 + 59836320*int(x**6/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11664*x**7 + 109296*x**6 + 69984*x**5 + 19440*x**4 + 139968*x**3 + 46656*x**2 + 46656),x)*x**3 + 19945440*int(x**6 /(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11664*x**7 + 109296*x**6 + 6998 4*x**5 + 19440*x**4 + 139968*x**3 + 46656*x**2 + 46656),x)*x**2 + 39890880 *int(x**6/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11664*x**7 + 109296...