Integrand size = 26, antiderivative size = 677 \[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {9 (-2)^{2/3}+\sqrt [3]{6} \left (9+\sqrt [3]{-3} 2^{2/3}\right ) x}{2916\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {9\ 2^{2/3}+\sqrt [3]{-1} 3^{2/3} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x}{13122\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (2-3 \sqrt [3]{2} 3^{2/3}\right ) x}{8748\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\sqrt [3]{-1} \left (3 (-3)^{2/3}-2^{2/3}\right ) \arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{486\ 6^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}+\frac {\left (3 (-3)^{2/3}+\sqrt [3]{-1} 2^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{486\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{486\ 6^{5/6} \left (1-\sqrt [3]{-1}\right )^2 \left (1+\sqrt [3]{-1}\right )^4 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{5832 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{5832 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{52488 \sqrt [3]{2} 3^{2/3}} \] Output:
1/5832*(9*(-2)^(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)+1/26244*(9*2 ^(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)+1/52488*(3*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)+1/2916*(-1)^(1/3)*(3*(-3)^(2/3)-2^(2/3))*ar ctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2))*6^(1/6)/ (1+(-1)^(1/3))^4/(4-3*(-3)^(2/3)*2^(1/3))^(3/2)+1/2916*(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)-1/2916*(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)+1/34992*(-1)^(1/6)*3^(5/6)*ln(6-3*(-3)^(1/3)*2^ (2/3)*x+x^2)*2^(2/3)/(1+(-1)^(1/3))^5-1/34992*I*ln(6+3*(-2)^(2/3)*3^(1/3)* x+x^2)*2^(2/3)*3^(5/6)/(1+(-1)^(1/3))^5+1/314928*ln(6+3*2^(2/3)*3^(1/3)*x+ x^2)*2^(2/3)*3^(1/3)
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 = 0.25 \[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {-96+108 x-64 x^2-72 x^3+73 x^4-3 x^5}{68364 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {108 \log (x-\text {$\#$1})-32 \log (x-\text {$\#$1}) \text {$\#$1}+108 \log (x-\text {$\#$1}) \text {$\#$1}^2-146 \log (x-\text {$\#$1}) \text {$\#$1}^3+3 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{410184} \] Input:
Integrate[x^6/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]
Output:
(-96 + 108*x - 64*x^2 - 72*x^3 + 73*x^4 - 3*x^5)/(68364*(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 & , (108*Log[x - #1] - 32*Log[x - #1]*#1 + 108*Log[x - #1]*#1^2 - 146*L og[x - #1]*#1^3 + 3*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/410184
Time = 3.25 (sec) , antiderivative size = 638, normalized size of antiderivative = 0.94, 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.
| \(\displaystyle \int \frac {x^6}{\left (x^6+18 x^4+324 x^3+108 x^2+216\right )^2} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1586874322944 \int \left (\frac {3^{5/6} \left (1+\sqrt [3]{-1}\right )-i \sqrt [3]{2} x}{4627325525704704\ 2^{2/3} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac {3 i 3^{5/6}-\left (\sqrt [3]{-2}+\sqrt [3]{2}\right ) x}{4627325525704704\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {\sqrt [3]{2} x+3 \sqrt [3]{3}}{41645929731342336\ 6^{2/3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}-\frac {(-1)^{2/3} \left (\sqrt [3]{3} x+\sqrt [3]{-2} \sqrt [3]{-1}\right )}{514147280633856 \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^4 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )^2}+\frac {2 \sqrt [3]{-1}-2^{2/3} \sqrt [3]{3} x}{4627325525704704\ 2^{2/3} \sqrt [3]{3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )^2}-\frac {\sqrt [3]{3} x+\sqrt [3]{2}}{4627325525704704 \sqrt [3]{3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1586874322944 \left (\frac {\left (3 (-3)^{2/3}+\sqrt [3]{-1} 2^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{6940988288557056\ 6^{5/6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {\left (\sqrt [3]{-1} 2^{2/3}+3\ 3^{2/3}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{771220920950784\ 6^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )^{3/2}}-\frac {\left (2^{2/3}-3\ 3^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{6940988288557056\ 6^{5/6} \left (3 \sqrt [3]{2} 3^{2/3}-4\right )^{3/2}}+\frac {\sqrt [3]{-1} \left (2+3 \sqrt [3]{-2} 3^{2/3}\right ) x+3\ 2^{2/3} \sqrt [3]{3}}{13881976577114112\ 2^{2/3} \sqrt [3]{3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {(-1)^{2/3} \left (6 \sqrt [3]{3}-\sqrt [3]{-2} \left (2 \sqrt [3]{-1}+3 \sqrt [3]{2} 3^{2/3}\right ) x\right )}{3084883683803136 \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {6 \sqrt [3]{3}-\left (2 \sqrt [3]{2}-3\ 6^{2/3}\right ) x}{27763953154228224 \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac {\sqrt [6]{-\frac {1}{3}} \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{9254651051409408 \sqrt [3]{2} \left (1+\sqrt [3]{-1}\right )^5}-\frac {i \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{9254651051409408 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{83291859462684672 \sqrt [3]{2} 3^{2/3}}\right )\) |
Input:
Int[x^6/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]
Output:
1586874322944*(((-1)^(2/3)*(6*3^(1/3) - (-2)^(1/3)*(2*(-1)^(1/3) + 3*2^(1/ 3)*3^(2/3))*x))/(3084883683803136*3^(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)) + (3*2^(2/3)*3^(1/3) + ( -1)^(1/3)*(2 + 3*(-2)^(1/3)*3^(2/3))*x)/(13881976577114112*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)) + (6*3^(1/ 3) - (2*2^(1/3) - 3*6^(2/3))*x)/(27763953154228224*3^(1/3)*(4 - 3*2^(1/3)* 3^(2/3))*(6 + 3*2^(2/3)*3^(1/3)*x + x^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 ))]])/(6940988288557056*6^(5/6)*(4 + 3*(-2)^(1/3)*3^(2/3))^(3/2)) - (((-1) ^(1/3)*2^(2/3) + 3*3^(2/3))*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sq rt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(771220920950784*6^(5/6)*(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))]])/(694098 8288557056*6^(5/6)*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) + ((-1/3)^(1/6)*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(9254651051409408*2^(1/3)*(1 + (-1)^(1/3)) ^5) - ((I/9254651051409408)*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]/(832918 59462684672*2^(1/3)*3^(2/3)))
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 = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.18
| method | result | size |
| default | \(\frac {-\frac {1}{22788} x^{5}+\frac {73}{68364} x^{4}-\frac {2}{1899} x^{3}-\frac {16}{17091} x^{2}+\frac {1}{633} x -\frac {8}{5697}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-3 \textit {\_R}^{4}+146 \textit {\_R}^{3}-108 \textit {\_R}^{2}+32 \textit {\_R} -108\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{410184}\) | \(122\) |
| risch | \(\frac {-\frac {1}{22788} x^{5}+\frac {73}{68364} x^{4}-\frac {2}{1899} x^{3}-\frac {16}{17091} x^{2}+\frac {1}{633} x -\frac {8}{5697}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-3 \textit {\_R}^{4}+146 \textit {\_R}^{3}-108 \textit {\_R}^{2}+32 \textit {\_R} -108\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{410184}\) | \(122\) |
Input:
int(x^6/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x,method=_RETURNVERBOSE)
Output:
(-1/22788*x^5+73/68364*x^4-2/1899*x^3-16/17091*x^2+1/633*x-8/5697)/(x^6+18 *x^4+324*x^3+108*x^2+216)+1/410184*sum((-3*_R^4+146*_R^3-108*_R^2+32*_R-10 8)/(_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. \[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^6/(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 = 0.17 \[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\operatorname {RootSum} {\left (3977704731623097128039995515166457856 t^{6} - 1010314319415295961050951680 t^{4} - 20168224477093957151232 t^{3} - 112582856818899648 t^{2} - 50648453064 t - 880007, \left ( t \mapsto t \log {\left (- \frac {273655567090018991570649941414395560986199688040644608 t^{5}}{49797855396139900267573395695} + \frac {11837008470196046085308646230764354292805044570112 t^{4}}{49797855396139900267573395695} - \frac {10570581900446717266374077482873315047787008 t^{3}}{49797855396139900267573395695} - \frac {1552547411569469872387563218792789323392 t^{2}}{49797855396139900267573395695} - \frac {12542923791159140826909003250295928 t}{49797855396139900267573395695} + x - \frac {23066533870320322410834348296}{49797855396139900267573395695} \right )} \right )\right )} + \frac {- 3 x^{5} + 73 x^{4} - 72 x^{3} - 64 x^{2} + 108 x - 96}{68364 x^{6} + 1230552 x^{4} + 22149936 x^{3} + 7383312 x^{2} + 14766624} \] Input:
integrate(x**6/(x**6+18*x**4+324*x**3+108*x**2+216)**2,x)
Output:
RootSum(3977704731623097128039995515166457856*_t**6 - 10103143194152959610 50951680*_t**4 - 20168224477093957151232*_t**3 - 112582856818899648*_t**2 - 50648453064*_t - 880007, Lambda(_t, _t*log(-2736555670900189915706499414 14395560986199688040644608*_t**5/49797855396139900267573395695 + 118370084 70196046085308646230764354292805044570112*_t**4/49797855396139900267573395 695 - 10570581900446717266374077482873315047787008*_t**3/49797855396139900 267573395695 - 1552547411569469872387563218792789323392*_t**2/497978553961 39900267573395695 - 12542923791159140826909003250295928*_t/497978553961399 00267573395695 + x - 23066533870320322410834348296/49797855396139900267573 395695))) + (-3*x**5 + 73*x**4 - 72*x**3 - 64*x**2 + 108*x - 96)/(68364*x* *6 + 1230552*x**4 + 22149936*x**3 + 7383312*x**2 + 14766624)
\[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{6}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \] Input:
integrate(x^6/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="maxima")
Output:
-1/68364*(3*x^5 - 73*x^4 + 72*x^3 + 64*x^2 - 108*x + 96)/(x^6 + 18*x^4 + 3 24*x^3 + 108*x^2 + 216) - 1/68364*integrate((3*x^4 - 146*x^3 + 108*x^2 - 3 2*x + 108)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
\[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{6}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \] Input:
integrate(x^6/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="giac")
Output:
integrate(x^6/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)^2, x)
Time = 22.29 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.57 \[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Too large to display} \] Input:
int(x^6/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216)^2,x)
Output:
symsum(log((7028852*root(z^6 - (60865*z^4)/239631364059408 - (15496909*z^3 )/3056398361930300326272 - (168169*z^2)/5941638415592503834272768 - (3971* z)/311864717157619341253309046784 - 880007/3977704731623097128039995515166 457856, z, k))/2628920529 - (1980083*x)/310470256633842 - (235710556*root( z^6 - (60865*z^4)/239631364059408 - (15496909*z^3)/3056398361930300326272 - (168169*z^2)/5941638415592503834272768 - (3971*z)/3118647171576193412533 09046784 - 880007/3977704731623097128039995515166457856, z, k)*x)/70980854 283 - (6628544*root(z^6 - (60865*z^4)/239631364059408 - (15496909*z^3)/305 6398361930300326272 - (168169*z^2)/5941638415592503834272768 - (3971*z)/31 1864717157619341253309046784 - 880007/397770473162309712803999551516645785 6, z, k)^2*x)/44521 - (141776759808*root(z^6 - (60865*z^4)/239631364059408 - (15496909*z^3)/3056398361930300326272 - (168169*z^2)/594163841559250383 4272768 - (3971*z)/311864717157619341253309046784 - 880007/397770473162309 7128039995515166457856, z, k)^3*x)/44521 + (183701926508544*root(z^6 - (60 865*z^4)/239631364059408 - (15496909*z^3)/3056398361930300326272 - (168169 *z^2)/5941638415592503834272768 - (3971*z)/311864717157619341253309046784 - 880007/3977704731623097128039995515166457856, z, k)^4*x)/211 - 694098828 8557056*root(z^6 - (60865*z^4)/239631364059408 - (15496909*z^3)/3056398361 930300326272 - (168169*z^2)/5941638415592503834272768 - (3971*z)/311864717 157619341253309046784 - 880007/3977704731623097128039995515166457856, z...
\[ \int \frac {x^6}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {too large to display} \] Input:
int(x^6/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x)
Output:
( - 3402*int(x**10/(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**6 - 61236*int(x**10/(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 - 1102248*int(x**10/(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 + 4665 6*x**2 + 46656),x)*x**3 - 367416*int(x**10/(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 - 734832*int(x**10/(x**12 + 36*x**10 + 648 *x**9 + 540*x**8 + 11664*x**7 + 109296*x**6 + 69984*x**5 + 19440*x**4 + 13 9968*x**3 + 46656*x**2 + 46656),x) + 6804*int(x**9/(x**12 + 36*x**10 + 648 *x**9 + 540*x**8 + 11664*x**7 + 109296*x**6 + 69984*x**5 + 19440*x**4 + 13 9968*x**3 + 46656*x**2 + 46656),x)*x**6 + 122472*int(x**9/(x**12 + 36*x**1 0 + 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 + 2204496*int(x**9/(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 + 734832*int(x**9/( 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 + 1469664*in t(x**9/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11664*x**7 + 109296*x*...