Integrand size = 26, antiderivative size = 682 \[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {\sqrt [3]{-\frac {1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{1944\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}+\frac {\sqrt [3]{-\frac {1}{3}} \left (4+(-2)^{2/3} \sqrt [3]{3} x\right )}{8748\ 2^{2/3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {4+2^{2/3} \sqrt [3]{3} x}{17496\ 2^{2/3} \sqrt [3]{3} \left (4-3 \sqrt [3]{2} 3^{2/3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}-\frac {\arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\arctan \left (\frac {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{4374 \sqrt {3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {i \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1458\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{4374 \sqrt {3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{8748 \sqrt {6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{39366\ 2^{5/6} \sqrt [6]{3} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}} \]
1/11664*(-1)^(1/3)*3^(2/3)*(4-(-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)+1/52488*(-1)^(1 /3)*3^(2/3)*(4+(-2)^(2/3)*3^(1/3)*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)+1/104976*(-4-2^(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)+1/1312 2*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2))/(8-9 *I*2^(1/3)*3^(1/6)+3*2^(1/3)*3^(2/3))^(3/2)*3^(1/2)-1/13122*arctan((3*(-2) ^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))/(8+9*I*2^(1/3)*3^(1/ 6)+3*2^(1/3)*3^(2/3))^(3/2)*3^(1/2)-1/52488*arctanh(2^(1/6)*(3*3^(1/3)+2^( 1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))/(-4+3*2^(1/3)*3^(2/3))^(3/2)*6^(1/2 )-1/26244*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/ 2))*2^(1/6)*3^(5/6)/(1+(-1)^(1/3))^4/(4-3*(-3)^(2/3)*2^(1/3))^(1/2)-1/8748 *I*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*2^( 1/6)*3^(1/3)/(1+(-1)^(1/3))^5/(4+3*(-2)^(1/3)*3^(2/3))^(1/2)-1/236196*arct anh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3))^(1/2))*2^(1/6)*3 ^(5/6)/(-4+3*2^(1/3)*3^(2/3))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.24 \[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {972-144 x+648 x^2+729 x^3-27 x^4+4 x^5}{615276 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}+\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {144 \log (x-\text {$\#$1})-324 \log (x-\text {$\#$1}) \text {$\#$1}+2043 \log (x-\text {$\#$1}) \text {$\#$1}^2-54 \log (x-\text {$\#$1}) \text {$\#$1}^3+4 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{3691656} \]
(972 - 144*x + 648*x^2 + 729*x^3 - 27*x^4 + 4*x^5)/(615276*(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 & , (144*Log[x - #1] - 324*Log[x - #1]*#1 + 2043*Log[x - #1]*#1^2 - 54*Log[x - #1]*#1^3 + 4*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1 ^5) & ]/3691656
Time = 1.62 (sec) , antiderivative size = 666, normalized size of antiderivative = 0.98, 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.
| \(\displaystyle \int \frac {x^5}{\left (x^6+18 x^4+324 x^3+108 x^2+216\right )^2} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1586874322944 \int \left (-\frac {\sqrt [3]{-\frac {1}{3}} x}{1542441841901568\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} x}{13881976577114112\ 2^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )^2}+\frac {x}{13881976577114112\ 2^{2/3} \sqrt [3]{3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )^2}+\frac {1}{4627325525704704 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac {i}{4627325525704704 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {1}{41645929731342336 \sqrt [3]{2} 3^{2/3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1586874322944 \left (-\frac {i \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 )}{2313662762852352\ 2^{5/6} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\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 )}{13881976577114112 \sqrt {6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}+\frac {\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 )}{6940988288557056 \sqrt {3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\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 )}{6940988288557056\ 2^{5/6} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^4 \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 (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{62468894597013504\ 2^{5/6} \sqrt [6]{3} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}-\frac {\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 )}{13881976577114112 \sqrt {6} \left (3 \sqrt [3]{2} 3^{2/3}-4\right )^{3/2}}+\frac {\sqrt [3]{-\frac {1}{3}} \left (4-\sqrt [3]{-3} 2^{2/3} x\right )}{3084883683803136\ 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]{-\frac {1}{3}} \left ((-2)^{2/3} \sqrt [3]{3} x+4\right )}{27763953154228224\ 2^{2/3} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac {2^{2/3} \sqrt [3]{3} x+4}{27763953154228224\ 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 )}\right )\) |
1586874322944*(((-1/3)^(1/3)*(4 - (-3)^(1/3)*2^(2/3)*x))/(3084883683803136 *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)) + ((-1/3)^(1/3)*(4 + (-2)^(2/3)*3^(1/3)*x))/(277639531542 28224*2^(2/3)*(4 + 3*(-2)^(1/3)*3^(2/3))*(6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2 )) - (4 + 2^(2/3)*3^(1/3)*x)/(27763953154228224*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)) - ArcTan[(3*(-2)^(2/3)*3^(1 /3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]]/(13881976577114112*Sqrt[6]* (4 + 3*(-2)^(1/3)*3^(2/3))^(3/2)) - ((I/2313662762852352)*ArcTan[(3*(-2)^( 2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(2^(5/6)*3^(2/3)* (1 + (-1)^(1/3))^5*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) - ArcTan[(2^(1/6)*(3*(- 3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]]/(69409882885570 56*2^(5/6)*3^(1/6)*(1 + (-1)^(1/3))^4*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + Ar cTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3) )]]/(6940988288557056*Sqrt[3]*(8 - (9*I)*2^(1/3)*3^(1/6) + 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))]]/(13881976577114112*Sqrt[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) )]]/(62468894597013504*2^(5/6)*3^(1/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]))
3.2.54.3.1 Defintions of rubi rules used
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 | \(\frac {\frac {1}{153819} x^{5}-\frac {1}{22788} x^{4}+\frac {1}{844} x^{3}+\frac {2}{1899} x^{2}-\frac {4}{17091} x +\frac {1}{633}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (4 \textit {\_R}^{4}-54 \textit {\_R}^{3}+2043 \textit {\_R}^{2}-324 \textit {\_R} +144\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{3691656}\) | \(122\) |
| risch | \(\frac {\frac {1}{153819} x^{5}-\frac {1}{22788} x^{4}+\frac {1}{844} x^{3}+\frac {2}{1899} x^{2}-\frac {4}{17091} x +\frac {1}{633}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (4 \textit {\_R}^{4}-54 \textit {\_R}^{3}+2043 \textit {\_R}^{2}-324 \textit {\_R} +144\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{3691656}\) | \(122\) |
(1/153819*x^5-1/22788*x^4+1/844*x^3+2/1899*x^2-4/17091*x+1/633)/(x^6+18*x^ 4+324*x^3+108*x^2+216)+1/3691656*sum((4*_R^4-54*_R^3+2043*_R^2-324*_R+144) /(_R^5+12*_R^3+162*_R^2+36*_R)*ln(x-_R),_R=RootOf(_Z^6+18*_Z^4+324*_Z^3+10 8*_Z^2+216))
Leaf count of result is larger than twice the leaf count of optimal. 1445 vs. \(2 (463) = 926\).
Time = 0.96 (sec) , antiderivative size = 1445, normalized size of antiderivative = 2.12 \[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Too large to display} \]
1/28041818976*(182304*x^5 - 1230552*x^4 + 33224904*x^3 + 422*sqrt(1/633)*( x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)*sqrt(5034474*18^(2/3) + 9367856*18 ^(1/3) + 44687457)*log(2/1982119441*sqrt(1/633)*(7238020557*(5034474*18^(2 /3) + 9367856*18^(1/3) + 44687457)^2 - 4479023748400406176979673*18^(2/3) - 8334306522507661258645112*18^(1/3) - 26862559811422885347120477)*sqrt(50 34474*18^(2/3) + 9367856*18^(1/3) + 44687457) - 7383041510/9393931*(503447 4*18^(2/3) + 9367856*18^(1/3) + 44687457)^2 + 247458158879850620*x + 51322 55454960803463351330/9393931*18^(2/3) + 9549802036377046040753520/9393931* 18^(1/3) + 27278928233033940032425830/9393931) - 422*sqrt(1/633)*(x^6 + 18 *x^4 + 324*x^3 + 108*x^2 + 216)*sqrt(5034474*18^(2/3) + 9367856*18^(1/3) + 44687457)*log(-2/1982119441*sqrt(1/633)*(7238020557*(5034474*18^(2/3) + 9 367856*18^(1/3) + 44687457)^2 - 4479023748400406176979673*18^(2/3) - 83343 06522507661258645112*18^(1/3) - 26862559811422885347120477)*sqrt(5034474*1 8^(2/3) + 9367856*18^(1/3) + 44687457) - 7383041510/9393931*(5034474*18^(2 /3) + 9367856*18^(1/3) + 44687457)^2 + 247458158879850620*x + 513225545496 0803463351330/9393931*18^(2/3) + 9549802036377046040753520/9393931*18^(1/3 ) + 27278928233033940032425830/9393931) - 9*sqrt(422)*(x^6 + 18*x^4 + 324* x^3 + 108*x^2 + 216)*sqrt(-20718*18^(2/3) + sqrt(-1/19683*(5034474*18^(2/3 ) + 9367856*18^(1/3) + 44687457)^2 + 22860116892*18^(2/3) + 3445478701088/ 81*18^(1/3) + 273974962699) - 9367856/243*18^(1/3) + 367798)*log(147660...
Time = 0.12 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.15 \[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\operatorname {RootSum} {\left (27493895104978847349012449000830556700672 t^{6} - 1318718189226950088862983192576 t^{4} + 12120917704776776448 t^{2} - 39753025, \left ( t \mapsto t \log {\left (\frac {947842259001288723909832054550209950242045952 t^{5}}{61864539719962655} - \frac {243458646817775607639654889480814592 t^{4}}{9811980923071} - \frac {41682556475067500431787310779667456 t^{3}}{61864539719962655} + \frac {12026877442664328616462272 t^{2}}{9811980923071} + \frac {216142618488859793668428 t}{61864539719962655} + x - \frac {308574300024117}{39247923692284} \right )} \right )\right )} + \frac {4 x^{5} - 27 x^{4} + 729 x^{3} + 648 x^{2} - 144 x + 972}{615276 x^{6} + 11074968 x^{4} + 199349424 x^{3} + 66449808 x^{2} + 132899616} \]
RootSum(27493895104978847349012449000830556700672*_t**6 - 1318718189226950 088862983192576*_t**4 + 12120917704776776448*_t**2 - 39753025, Lambda(_t, _t*log(947842259001288723909832054550209950242045952*_t**5/618645397199626 55 - 243458646817775607639654889480814592*_t**4/9811980923071 - 4168255647 5067500431787310779667456*_t**3/61864539719962655 + 1202687744266432861646 2272*_t**2/9811980923071 + 216142618488859793668428*_t/61864539719962655 + x - 308574300024117/39247923692284))) + (4*x**5 - 27*x**4 + 729*x**3 + 64 8*x**2 - 144*x + 972)/(615276*x**6 + 11074968*x**4 + 199349424*x**3 + 6644 9808*x**2 + 132899616)
\[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{5}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
1/615276*(4*x^5 - 27*x^4 + 729*x^3 + 648*x^2 - 144*x + 972)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216) + 1/615276*integrate((4*x^4 - 54*x^3 + 2043*x^2 - 324*x + 144)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
\[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{5}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
Time = 0.19 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.44 \[ \int \frac {x^5}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\left (\sum _{k=1}^6\ln \left (-\frac {4477969\,\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}{189282278088}+\frac {6305\,x}{4967524106141472}-\frac {\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )\,x\,16340881}{5110621508376}-\frac {{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^2\,x\,43348696}{10818603}-\frac {{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^3\,x\,65333687616}{44521}-\frac {{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^4\,x\,40024496812032}{211}-{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^5\,x\,6940988288557056+\frac {5943884\,{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^2}{400689}+\frac {224442467136\,{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^3}{44521}-\frac {137087493272064\,{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^4}{211}-168897381688221696\,{\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )}^5-\frac {13082875}{178830867821092992}\right )\,\mathrm {root}\left (z^6-\frac {183899\,z^4}{3834101824950528}+\frac {6209\,z^2}{14083883651774823903461376}-\frac {39753025}{27493895104978847349012449000830556700672},z,k\right )\right )+\frac {\frac {x^5}{153819}-\frac {x^4}{22788}+\frac {x^3}{844}+\frac {2\,x^2}{1899}-\frac {4\,x}{17091}+\frac {1}{633}}{x^6+18\,x^4+324\,x^3+108\,x^2+216} \]
symsum(log((6305*x)/4967524106141472 - (4477969*root(z^6 - (183899*z^4)/38 34101824950528 + (6209*z^2)/14083883651774823903461376 - 39753025/27493895 104978847349012449000830556700672, z, k))/189282278088 - (16340881*root(z^ 6 - (183899*z^4)/3834101824950528 + (6209*z^2)/14083883651774823903461376 - 39753025/27493895104978847349012449000830556700672, z, k)*x)/51106215083 76 - (43348696*root(z^6 - (183899*z^4)/3834101824950528 + (6209*z^2)/14083 883651774823903461376 - 39753025/27493895104978847349012449000830556700672 , z, k)^2*x)/10818603 - (65333687616*root(z^6 - (183899*z^4)/3834101824950 528 + (6209*z^2)/14083883651774823903461376 - 39753025/2749389510497884734 9012449000830556700672, z, k)^3*x)/44521 - (40024496812032*root(z^6 - (183 899*z^4)/3834101824950528 + (6209*z^2)/14083883651774823903461376 - 397530 25/27493895104978847349012449000830556700672, z, k)^4*x)/211 - 69409882885 57056*root(z^6 - (183899*z^4)/3834101824950528 + (6209*z^2)/14083883651774 823903461376 - 39753025/27493895104978847349012449000830556700672, z, k)^5 *x + (5943884*root(z^6 - (183899*z^4)/3834101824950528 + (6209*z^2)/140838 83651774823903461376 - 39753025/27493895104978847349012449000830556700672, z, k)^2)/400689 + (224442467136*root(z^6 - (183899*z^4)/3834101824950528 + (6209*z^2)/14083883651774823903461376 - 39753025/27493895104978847349012 449000830556700672, z, k)^3)/44521 - (137087493272064*root(z^6 - (183899*z ^4)/3834101824950528 + (6209*z^2)/14083883651774823903461376 - 39753025...