Integrand size = 26, antiderivative size = 1001 \[ \int \frac {x^7}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx =\text {Too large to display} \] Output:
-1/1944*(4*(-1)^(1/3)*3^(2/3)+18*6^(1/3)-9*((-2)^(2/3)+2*(-1)^(1/3)*3^(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)-1/4374*((-6)^(1/3)*(9*(-2)^(1/3)+2*3^(1/3))-9*(1+(-2)^(1/3)*3 ^(2/3))*x)/(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)+1/17496*(4-6*2^(1/3)*3^(2/3)-3*(6-2^(2/3)*3^(1/3))*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)+1/5832*(9*I+3^(1/3 )*(2*I*2^(2/3)-9*3^(1/6)+2*2^(2/3)*3^(1/2)))*arctan((3*(-3)^(1/3)*2^(2/3)- 2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2))/(1+(-1)^(1/3))^5/(8-6*(-3)^(2/3)*2^ (1/3))^(1/2)+1/324*(1+(-2)^(1/3)*3^(2/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x )/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*6^(1/2)/(1-(-1)^(1/3))^2/(1+(-1)^(1/3) )^4/(4+3*(-2)^(1/3)*3^(2/3))^(3/2)-1/5832*(9*I-3^(1/3)*(4*I*2^(2/3)+9*3^(1 /6)))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))/ (1+(-1)^(1/3))^5/(8+6*(-2)^(1/3)*3^(2/3))^(1/2)-1/324*(-1)^(1/3)*((-3)^(1/ 3)+3*2^(1/3))*arctan(2^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/(12-9*(-3)^(2/3)*2^( 1/3))^(1/2))*2^(1/2)*3^(1/6)/(1+(-1)^(1/3))^4/(4-3*(-3)^(2/3)*2^(1/3))^(3/ 2)+1/324*(1-2^(1/3)*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^(1/2)/(1-(-1)^(1/3))^2/(1+(-1)^(1/3))^4/(-4+3*2^ (1/3)*3^(2/3))^(3/2)+1/78732*(2*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))*3^(5/6)/(-8+6*2^(1/3)*3^(2/3 ))^(1/2)+1/3888*I*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(1/3)*3^(1/6)/(1+(...
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 \[ \int \frac {x^7}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {648-96 x+432 x^2+908 x^3-18 x^4+73 x^5}{68364 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}+\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {96 \log (x-\text {$\#$1})-216 \log (x-\text {$\#$1}) \text {$\#$1}+96 \log (x-\text {$\#$1}) \text {$\#$1}^2-36 \log (x-\text {$\#$1}) \text {$\#$1}^3+73 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{410184} \] Input:
Integrate[x^7/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]
Output:
(648 - 96*x + 432*x^2 + 908*x^3 - 18*x^4 + 73*x^5)/(68364*(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 & , (96*Log[x - #1] - 216*Log[x - #1]*#1 + 96*Log[x - #1]*#1^2 - 36*L og[x - #1]*#1^3 + 73*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/410184
Time = 4.48 (sec) , antiderivative size = 954, normalized size of antiderivative = 0.95, 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.
| \(\displaystyle \int \frac {x^7}{\left (x^6+18 x^4+324 x^3+108 x^2+216\right )^2} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1586874322944 \int \left (\frac {3 i \sqrt [3]{2} \sqrt [6]{3} x-i 2^{2/3} 3^{5/6}-9 i \sqrt {3}-3\ 2^{2/3} \sqrt [3]{3}+27}{9254651051409408 \left (1+\sqrt [3]{-1}\right )^5 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {2^{2/3} \left (27-9 i \sqrt {3}+2 i 2^{2/3} 3^{5/6}\right )-3 \sqrt [6]{3} \left (i+\sqrt {3}\right ) x}{9254651051409408\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac {\sqrt [3]{2} 3^{2/3} x+2 \left (9+2^{2/3} \sqrt [3]{3}\right )}{83291859462684672 \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac {\sqrt [3]{3} \left (\sqrt [3]{-3}+9 \sqrt [3]{2}\right ) x+9 (-2)^{2/3}}{771220920950784\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )^2}+\frac {\sqrt [3]{-1} 3^{2/3} \left (1+3 \sqrt [3]{-2} 3^{2/3}\right ) x+9\ 2^{2/3}}{6940988288557056\ 2^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )^2}+\frac {3\ 2^{2/3} \sqrt [3]{3}-\left (1-3 \sqrt [3]{2} 3^{2/3}\right ) x}{2313662762852352\ 2^{2/3} \sqrt [3]{3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1586874322944 \left (\frac {2 \left (2-3 \sqrt [3]{2} 3^{2/3}\right )-3 \left (6-2^{2/3} \sqrt [3]{3}\right ) x}{4627325525704704\ 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 {\left (9 i-\sqrt [3]{3} \left (4 i 2^{2/3}+9 \sqrt [6]{3}\right )\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 )}{9254651051409408 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}+\frac {\left (1+\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 )}{771220920950784 \sqrt {6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}+\frac {\left (9 i+\sqrt [3]{3} \left (2 i 2^{2/3}-9 \sqrt [6]{3}+2\ 2^{2/3} \sqrt {3}\right )\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 )}{9254651051409408 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\frac {\sqrt [3]{-1} \left (\sqrt [3]{-3}+3 \sqrt [3]{2}\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 )}{85691213438976 \sqrt {2} 3^{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^{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 )}{41645929731342336 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}+\frac {\left (1-\sqrt [3]{2} 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 )}{771220920950784 \sqrt {6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {i \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{1028294561267712\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\left (i+\sqrt {3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{2056589122535424\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}-\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{27763953154228224\ 2^{2/3} \sqrt [3]{3}}-\frac {2 \left (2 \sqrt [3]{-1} 3^{2/3}+9 \sqrt [3]{6}\right )-9 \left ((-2)^{2/3}+2 \sqrt [3]{-1} 3^{2/3}\right ) x}{1542441841901568\ 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 {\sqrt [3]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )-9 \left (1+\sqrt [3]{-2} 3^{2/3}\right ) x}{13881976577114112 \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^7/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]
Output:
1586874322944*(-1/1542441841901568*(2*(2*(-1)^(1/3)*3^(2/3) + 9*6^(1/3)) - 9*((-2)^(2/3) + 2*(-1)^(1/3)*3^(2/3))*x)/(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)) - ((-6)^(1/3)*( 9*(-2)^(1/3) + 2*3^(1/3)) - 9*(1 + (-2)^(1/3)*3^(2/3))*x)/(138819765771141 12*(4 + 3*(-2)^(1/3)*3^(2/3))*(6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2)) + (2*(2 - 3*2^(1/3)*3^(2/3)) - 3*(6 - 2^(2/3)*3^(1/3))*x)/(4627325525704704*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 + (-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))]])/(771220920950784*Sqrt[6]*(4 + 3*(-2)^(1/3)*3^(2/3))^(3 /2)) - ((9*I - 3^(1/3)*((4*I)*2^(2/3) + 9*3^(1/6)))*ArcTan[(3*(-2)^(2/3)*3 ^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(9254651051409408*(1 + (-1)^(1/3))^5*Sqrt[2*(4 + 3*(-2)^(1/3)*3^(2/3))]) - ((-1)^(1/3)*((-3)^(1/3 ) + 3*2^(1/3))*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*( -3)^(2/3)*2^(1/3))]])/(85691213438976*Sqrt[2]*3^(5/6)*(1 + (-1)^(1/3))^4*( 4 - 3*(-3)^(2/3)*2^(1/3))^(3/2)) + ((9*I + 3^(1/3)*((2*I)*2^(2/3) - 9*3^(1 /6) + 2*2^(2/3)*Sqrt[3]))*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt [3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(9254651051409408*(1 + (-1)^(1/3))^5*Sqrt [2*(4 - 3*(-3)^(2/3)*2^(1/3))]) + ((1 - 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))]])/(7712209209507 84*Sqrt[6]*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) + ((2*2^(2/3) + 3*3^(2/3))*A...
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 = 0.12
| method | result | size |
| default | \(\frac {\frac {73}{68364} x^{5}-\frac {1}{3798} x^{4}+\frac {227}{17091} x^{3}+\frac {4}{633} x^{2}-\frac {8}{5697} x +\frac {2}{211}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (73 \textit {\_R}^{4}-36 \textit {\_R}^{3}+96 \textit {\_R}^{2}-216 \textit {\_R} +96\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{410184}\) | \(122\) |
| risch | \(\frac {\frac {73}{68364} x^{5}-\frac {1}{3798} x^{4}+\frac {227}{17091} x^{3}+\frac {4}{633} x^{2}-\frac {8}{5697} x +\frac {2}{211}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (73 \textit {\_R}^{4}-36 \textit {\_R}^{3}+96 \textit {\_R}^{2}-216 \textit {\_R} +96\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{410184}\) | \(122\) |
Input:
int(x^7/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x,method=_RETURNVERBOSE)
Output:
(73/68364*x^5-1/3798*x^4+227/17091*x^3+4/633*x^2-8/5697*x+2/211)/(x^6+18*x ^4+324*x^3+108*x^2+216)+1/410184*sum((73*_R^4-36*_R^3+96*_R^2-216*_R+96)/( _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. \[ \int \frac {x^7}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Timed out} \] Input:
integrate(x^7/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="fricas")
Output:
Timed out
Time = 0.23 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.11 \[ \int \frac {x^7}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\operatorname {RootSum} {\left (589289589870088463413332668913549312 t^{6} - 539640290266075248405737472 t^{4} + 92182638168509682392064 t^{3} - 553241442069170496 t^{2} - 3759837842016 t - 7197829, \left ( t \mapsto t \log {\left (\frac {42996027639727447714003743305160746111018438501025999323136 t^{5}}{154206009791052044490694380303237521} - \frac {42584766259508194684689715474422251405157209835847680 t^{4}}{154206009791052044490694380303237521} - \frac {37512446128849588150108369449323754078317341082112 t^{3}}{154206009791052044490694380303237521} + \frac {7152037594021675267638890715531672481920222144 t^{2}}{154206009791052044490694380303237521} - \frac {44227546998835297723830291794974310524032 t}{154206009791052044490694380303237521} + x - \frac {174573349036676047734132569583024855}{154206009791052044490694380303237521} \right )} \right )\right )} + \frac {73 x^{5} - 18 x^{4} + 908 x^{3} + 432 x^{2} - 96 x + 648}{68364 x^{6} + 1230552 x^{4} + 22149936 x^{3} + 7383312 x^{2} + 14766624} \] Input:
integrate(x**7/(x**6+18*x**4+324*x**3+108*x**2+216)**2,x)
Output:
RootSum(589289589870088463413332668913549312*_t**6 - 539640290266075248405 737472*_t**4 + 92182638168509682392064*_t**3 - 553241442069170496*_t**2 - 3759837842016*_t - 7197829, Lambda(_t, _t*log(4299602763972744771400374330 5160746111018438501025999323136*_t**5/154206009791052044490694380303237521 - 42584766259508194684689715474422251405157209835847680*_t**4/15420600979 1052044490694380303237521 - 3751244612884958815010836944932375407831734108 2112*_t**3/154206009791052044490694380303237521 + 715203759402167526763889 0715531672481920222144*_t**2/154206009791052044490694380303237521 - 442275 46998835297723830291794974310524032*_t/15420600979105204449069438030323752 1 + x - 174573349036676047734132569583024855/15420600979105204449069438030 3237521))) + (73*x**5 - 18*x**4 + 908*x**3 + 432*x**2 - 96*x + 648)/(68364 *x**6 + 1230552*x**4 + 22149936*x**3 + 7383312*x**2 + 14766624)
\[ \int \frac {x^7}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{7}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \] Input:
integrate(x^7/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="maxima")
Output:
1/68364*(73*x^5 - 18*x^4 + 908*x^3 + 432*x^2 - 96*x + 648)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216) + 1/68364*integrate((73*x^4 - 36*x^3 + 96*x^2 - 216*x + 96)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
\[ \int \frac {x^7}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{7}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \] Input:
integrate(x^7/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="giac")
Output:
integrate(x^7/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)^2, x)
Time = 22.18 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.39 \[ \int \frac {x^7}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Too large to display} \] Input:
int(x^7/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216)^2,x)
Output:
symsum(log((8336932*root(z^6 - (292589*z^4)/319508485412544 + (11805253*z^ 3)/75466626220501242624 - (2479189*z^2)/2640728184707779481899008 - (19897 87*z)/311864717157619341253309046784 - 7197829/589289589870088463413332668 913549312, z, k))/97367427 - (480227*x)/851770251396 - (759164282*root(z^6 - (292589*z^4)/319508485412544 + (11805253*z^3)/75466626220501242624 - (2 479189*z^2)/2640728184707779481899008 - (1989787*z)/3118647171576193412533 09046784 - 7197829/589289589870088463413332668913549312, z, k)*x)/78867615 87 - (207565888*root(z^6 - (292589*z^4)/319508485412544 + (11805253*z^3)/7 5466626220501242624 - (2479189*z^2)/2640728184707779481899008 - (1989787*z )/311864717157619341253309046784 - 7197829/5892895898700884634133326689135 49312, z, k)^2*x)/400689 - (108430970112*root(z^6 - (292589*z^4)/319508485 412544 + (11805253*z^3)/75466626220501242624 - (2479189*z^2)/2640728184707 779481899008 - (1989787*z)/311864717157619341253309046784 - 7197829/589289 589870088463413332668913549312, z, k)^3*x)/44521 - (147138513610752*root(z ^6 - (292589*z^4)/319508485412544 + (11805253*z^3)/75466626220501242624 - (2479189*z^2)/2640728184707779481899008 - (1989787*z)/31186471715761934125 3309046784 - 7197829/589289589870088463413332668913549312, z, k)^4*x)/211 - 6940988288557056*root(z^6 - (292589*z^4)/319508485412544 + (11805253*z^3 )/75466626220501242624 - (2479189*z^2)/2640728184707779481899008 - (198978 7*z)/311864717157619341253309046784 - 7197829/5892895898700884634133326...
\[ \int \frac {x^7}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {too large to display} \] Input:
int(x^7/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x)
Output:
( - 2916*int(x**8/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11664*x**7 + 1 09296*x**6 + 69984*x**5 + 19440*x**4 + 139968*x**3 + 46656*x**2 + 46656),x )*x**6 - 52488*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 + 46 656),x)*x**4 - 944784*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 - 314928*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 + 4 6656*x**2 + 46656),x)*x**2 - 629856*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) - 125496*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**6 - 2258928*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 - 40660704*int(x**6/(x**12 + 3 6*x**10 + 648*x**9 + 540*x**8 + 11664*x**7 + 109296*x**6 + 69984*x**5 + 19 440*x**4 + 139968*x**3 + 46656*x**2 + 46656),x)*x**3 - 13553568*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**2 - 27107136*i nt(x**6/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11664*x**7 + 109296*x...