Integrand size = 26, antiderivative size = 873 \[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {\sqrt [3]{-6} \left (2 \sqrt [3]{-3}+9 \sqrt [3]{2}\right )-3 x}{157464 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right ) \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}-\frac {\sqrt [3]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )+3 x}{157464 \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 {2 \sqrt [3]{2}-3\ 6^{2/3}-\sqrt [3]{3} x}{104976 \left (9 \sqrt [3]{2}-4 \sqrt [3]{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 )}{26244 \sqrt {3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+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 {3 \sqrt [3]{-3} 2^{2/3}-2 x}{\sqrt {6 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}-\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 )}{26244 \sqrt {3} \left (8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (9 i+\sqrt [3]{3} \left (4 i 2^{2/3}-9 \sqrt [6]{3}\right )\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 )}{209952 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}-\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 )}{52488 \sqrt {6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}+\frac {\left (2\ 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 )}{944784 \sqrt [6]{3} \sqrt {2 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}-\frac {i \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{23328\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\left (i+\sqrt {3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{46656\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^5}+\frac {\log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{629856\ 2^{2/3} \sqrt [3]{3}} \]
1/157464*((-6)^(1/3)*(2*(-3)^(1/3)+9*2^(1/3))-3*x)/(8-9*I*2^(1/3)*3^(1/6)+ 3*2^(1/3)*3^(2/3))/(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)+1/157464*(-(-6)^(1/3)*(9 *(-2)^(1/3)+2*3^(1/3))-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)+1/104976*(-2*2^(1/3)+3*6^(2/3)+3^(1/3)*x)/(9*2^ (1/3)-4*3^(1/3))/(6+3*2^(2/3)*3^(1/3)*x+x^2)-1/139968*I*ln(6-3*(-3)^(1/3)* 2^(2/3)*x+x^2)*2^(1/3)*3^(1/6)/(1+(-1)^(1/3))^5+1/3779136*ln(6+3*2^(2/3)*3 ^(1/3)*x+x^2)*2^(1/3)*3^(2/3)+1/78732*arctan((3*(-3)^(1/3)*2^(2/3)-2*x)/(2 4-18*(-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)-1/78732*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/ 279936*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)*(3^(1/2)+I)*2^(1/3)*3^(1/6)/(1+(-1 )^(1/3))^5-1/314928*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)-1/209952*arctan((3*(-3 )^(1/3)*2^(2/3)-2*x)/(24-18*(-3)^(2/3)*2^(1/3))^(1/2))*(9*I-3^(1/3)*(2*I*2 ^(2/3)+9*3^(1/6)+2*2^(2/3)*3^(1/2)))/(1+(-1)^(1/3))^5/(8-6*(-3)^(2/3)*2^(1 /3))^(1/2)+1/209952*(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)+1/2834352*(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)
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 = 0.19 \[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {972-3942 x+648 x^2+96 x^3-27 x^4+4 x^5}{3691656 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}+\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {1971 \log (x-\text {$\#$1})-162 \log (x-\text {$\#$1}) \text {$\#$1}+72 \log (x-\text {$\#$1}) \text {$\#$1}^2-27 \log (x-\text {$\#$1}) \text {$\#$1}^3+2 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{11074968} \]
(972 - 3942*x + 648*x^2 + 96*x^3 - 27*x^4 + 4*x^5)/(3691656*(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 & , (1971*Log[x - #1] - 162*Log[x - #1]*#1 + 72*Log[x - #1]*#1^2 - 27*Log[x - #1]*#1^3 + 2*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 + #1 ^5) & ]/11074968
Time = 2.39 (sec) , antiderivative size = 855, 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.
| \(\displaystyle \int \frac {x^3}{\left (x^6+18 x^4+324 x^3+108 x^2+216\right )^2} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1586874322944 \int \left (-\frac {x+3\ 2^{2/3} \sqrt [3]{3}}{83291859462684672\ 2^{2/3} \sqrt [3]{3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )^2}-\frac {3 i \sqrt [3]{2} \sqrt [6]{3} x+i 2^{2/3} 3^{5/6}-9 i \sqrt {3}+3\ 2^{2/3} \sqrt [3]{3}+27}{333167437850738688 \left (1+\sqrt [3]{-1}\right )^5 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac {2 \left (27-i \left (9 \sqrt {3}+2\ 2^{2/3} 3^{5/6}\right )\right )-3 \sqrt [3]{2} \sqrt [6]{3} \left (i+\sqrt {3}\right ) x}{666334875701477376 \left (1+\sqrt [3]{-1}\right )^5 \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {3^{2/3} x+2^{2/3} \left (9-2^{2/3} \sqrt [3]{3}\right )}{1499253470328324096\ 2^{2/3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}-\frac {9 (-2)^{2/3}-\sqrt [3]{-1} 3^{2/3} x}{27763953154228224\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )^2}-\frac {9\ 2^{2/3}-\sqrt [3]{-1} 3^{2/3} x}{249875578388054016\ 2^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1586874322944 \left (-\frac {3 x+\sqrt [3]{-6} \left (9 \sqrt [3]{-2}+2 \sqrt [3]{3}\right )}{499751156776108032 \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 )}{333167437850738688 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}-\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 )}{83291859462684672 \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 )}{333167437850738688 \left (1+\sqrt [3]{-1}\right )^5 \sqrt {2 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}+\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 )}{41645929731342336 \sqrt {3} \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\sqrt {-4+3 \sqrt [3]{2} 3^{2/3}} \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 )}{1499253470328324096\ 2^{5/6} \sqrt [6]{3}}-\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 )}{83291859462684672 \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 )}{37018604205637632\ 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 )}{74037208411275264\ 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 )}{999502313552216064\ 2^{2/3} \sqrt [3]{3}}-\frac {3 (-2)^{2/3} x+2 \left (2 \sqrt [3]{-1} 3^{2/3}+9 \sqrt [3]{6}\right )}{55527906308456448\ 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]{3} x-3\ 6^{2/3}+2 \sqrt [3]{2}}{166583718925369344 \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*(-1/55527906308456448*(2*(2*(-1)^(1/3)*3^(2/3) + 9*6^(1/3)) + 3*(-2)^(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) ) + 3*x)/(499751156776108032*(4 + 3*(-2)^(1/3)*3^(2/3))*(6 + 3*(-2)^(2/3)* 3^(1/3)*x + x^2)) + (2*2^(1/3) - 3*6^(2/3) - 3^(1/3)*x)/(16658371892536934 4*3^(1/3)*(4 - 3*2^(1/3)*3^(2/3))*(6 + 3*2^(2/3)*3^(1/3)*x + x^2)) - ArcTa n[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]]/(832918 59462684672*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))]])/(333167437850738688*(1 + (-1)^(1/3))^5*Sqrt[2*(4 + 3*(-2)^(1/3)*3^(2/3))]) + ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x))/S qrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]]/(41645929731342336*Sqrt[3]*(8 - (9*I)*2 ^(1/3)*3^(1/6) + 3*2^(1/3)*3^(2/3))^(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))]])/(333167437850738688*(1 + (-1)^( 1/3))^5*Sqrt[2*(4 - 3*(-3)^(2/3)*2^(1/3))]) - ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]]/(83291859462684672*Sqrt[6] *(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) - (Sqrt[-4 + 3*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))]])/(14992 53470328324096*2^(5/6)*3^(1/6)) - ((I/37018604205637632)*Log[6 - 3*(-3)...
3.2.56.3.1 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 = 0.14
| method | result | size |
| default | \(\frac {\frac {1}{922914} x^{5}-\frac {1}{136728} x^{4}+\frac {4}{153819} x^{3}+\frac {1}{5697} x^{2}-\frac {73}{68364} x +\frac {1}{3798}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}-27 \textit {\_R}^{3}+72 \textit {\_R}^{2}-162 \textit {\_R} +1971\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{11074968}\) | \(122\) |
| risch | \(\frac {\frac {1}{922914} x^{5}-\frac {1}{136728} x^{4}+\frac {4}{153819} x^{3}+\frac {1}{5697} x^{2}-\frac {73}{68364} x +\frac {1}{3798}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}-27 \textit {\_R}^{3}+72 \textit {\_R}^{2}-162 \textit {\_R} +1971\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{11074968}\) | \(122\) |
(1/922914*x^5-1/136728*x^4+4/153819*x^3+1/5697*x^2-73/68364*x+1/3798)/(x^6 +18*x^4+324*x^3+108*x^2+216)+1/11074968*sum((2*_R^4-27*_R^3+72*_R^2-162*_R +1971)/(_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. \[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.13 \[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\operatorname {RootSum} {\left (1282755170017893101915524820582750453426552832 t^{6} - 906388465775544244426251149770752 t^{4} - 4300873166389987741684137984 t^{3} - 717000908921644962816 t^{2} + 135354162312576 t - 7197829, \left ( t \mapsto t \log {\left (\frac {17257935592810449901409556597891882995604001083339368041361480613888 t^{5}}{154206009791052044490694380303237521} + \frac {2389607400620985524376358853572652207181956324560587684052992 t^{4}}{154206009791052044490694380303237521} - \frac {12286072160883283930711715948878260078996992193488388096 t^{3}}{154206009791052044490694380303237521} - \frac {59490553573959173161125496013527909754156558410752 t^{2}}{154206009791052044490694380303237521} - \frac {17520149679836691112367064197713753004827200 t}{154206009791052044490694380303237521} + x + \frac {766422988707229615055855287040887332}{154206009791052044490694380303237521} \right )} \right )\right )} + \frac {4 x^{5} - 27 x^{4} + 96 x^{3} + 648 x^{2} - 3942 x + 972}{3691656 x^{6} + 66449808 x^{4} + 1196096544 x^{3} + 398698848 x^{2} + 797397696} \]
RootSum(1282755170017893101915524820582750453426552832*_t**6 - 90638846577 5544244426251149770752*_t**4 - 4300873166389987741684137984*_t**3 - 717000 908921644962816*_t**2 + 135354162312576*_t - 7197829, Lambda(_t, _t*log(17 257935592810449901409556597891882995604001083339368041361480613888*_t**5/1 54206009791052044490694380303237521 + 238960740062098552437635885357265220 7181956324560587684052992*_t**4/154206009791052044490694380303237521 - 122 86072160883283930711715948878260078996992193488388096*_t**3/15420600979105 2044490694380303237521 - 5949055357395917316112549601352790975415655841075 2*_t**2/154206009791052044490694380303237521 - 175201496798366911123670641 97713753004827200*_t/154206009791052044490694380303237521 + x + 7664229887 07229615055855287040887332/154206009791052044490694380303237521))) + (4*x* *5 - 27*x**4 + 96*x**3 + 648*x**2 - 3942*x + 972)/(3691656*x**6 + 66449808 *x**4 + 1196096544*x**3 + 398698848*x**2 + 797397696)
\[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{3}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
1/3691656*(4*x^5 - 27*x^4 + 96*x^3 + 648*x^2 - 3942*x + 972)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216) + 1/1845828*integrate((2*x^4 - 27*x^3 + 72*x^2 - 162*x + 1971)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
\[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{3}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
Time = 9.31 (sec) , antiderivative size = 387, normalized size of antiderivative = 0.44 \[ \int \frac {x^3}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Too large to display} \]
symsum(log((11*x)/603554178896188848 - (14059*root(z^6 - (292589*z^4)/4140 82997094657024 - (11805253*z^3)/3520970912943705975865344 - (2479189*z^2)/ 4435409310686141742269284220928 + (1989787*z)/1885726687584283082922664540 5233577984 - 7197829/1282755170017893101915524820582750453426552832, z, k) )/30663729050256 - (5658601*root(z^6 - (292589*z^4)/414082997094657024 - ( 11805253*z^3)/3520970912943705975865344 - (2479189*z^2)/443540931068614174 2269284220928 + (1989787*z)/18857266875842830829226645405233577984 - 71978 29/1282755170017893101915524820582750453426552832, z, k)*x)/66233654748552 96 + (6603523*root(z^6 - (292589*z^4)/414082997094657024 - (11805253*z^3)/ 3520970912943705975865344 - (2479189*z^2)/4435409310686141742269284220928 + (1989787*z)/18857266875842830829226645405233577984 - 7197829/12827551700 17893101915524820582750453426552832, z, k)^2*x)/584204562 - (1762321104*ro ot(z^6 - (292589*z^4)/414082997094657024 - (11805253*z^3)/3520970912943705 975865344 - (2479189*z^2)/4435409310686141742269284220928 + (1989787*z)/18 857266875842830829226645405233577984 - 7197829/128275517001789310191552482 0582750453426552832, z, k)^3*x)/44521 - (59633904436992*root(z^6 - (292589 *z^4)/414082997094657024 - (11805253*z^3)/3520970912943705975865344 - (247 9189*z^2)/4435409310686141742269284220928 + (1989787*z)/188572668758428308 29226645405233577984 - 7197829/1282755170017893101915524820582750453426552 832, z, k)^4*x)/211 - 6940988288557056*root(z^6 - (292589*z^4)/41408299...