Integrand size = 26, antiderivative size = 850 \[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {\sqrt [3]{-\frac {1}{3}} \left (3 \sqrt [3]{-3} 2^{2/3}-2 x\right )}{5832\ 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 (3 (-2)^{2/3} \sqrt [3]{3}+2 x\right )}{26244\ 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 \sqrt [3]{3}+\sqrt [3]{2} x}{52488 \left (9 \sqrt [3]{2}-4 \sqrt [3]{3}\right ) \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}+\frac {\sqrt [3]{-1} \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 )}{729\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \left (8-9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\sqrt [3]{-1} \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 )}{2916 \sqrt [6]{2} 3^{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 (i+\sqrt {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 )}{11664 \sqrt [6]{2} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\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 )}{5832 \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 (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{26244 \sqrt [6]{2} 3^{5/6} \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 )}{52488 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {\log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{34992 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac {i \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{34992 \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 )}{314928 \sqrt [3]{2} 3^{2/3}} \]
1/34992*(-1)^(1/3)*3^(2/3)*(3*(-3)^(1/3)*2^(2/3)-2*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)-1/157464*(-1) ^(1/3)*3^(2/3)*(3*(-2)^(2/3)*3^(1/3)+2*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*3^(1/3)-2^(1/ 3)*x)/(9*2^(1/3)-4*3^(1/3))/(6+3*2^(2/3)*3^(1/3)*x+x^2)+1/4374*(-1)^(1/3)* 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) -1/17496*(-1)^(1/3)*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^ (2/3))^(1/2))*2^(5/6)*3^(1/6)/(1-(-1)^(1/3))^2/(1+(-1)^(1/3))^4/(4+3*(-2)^ (1/3)*3^(2/3))^(3/2)+1/157464*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))^(3/2)-1/20 9952*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(2/3)*3^(1/3)/(1+(-1)^(1/3))^4+1/2 09952*I*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2)*2^(2/3)*3^(5/6)/(1+(-1)^(1/3))^5- 1/1889568*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)*2^(2/3)*3^(1/3)-1/34992*I*arctan(2 ^(1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/(12-9*(-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)-1/69984*arctan((3*( -2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*(3^(1/2)+I)*2^(5/ 6)*3^(2/3)/(1+(-1)^(1/3))^5/(4+3*(-2)^(1/3)*3^(2/3))^(1/2)+1/314928*arctan h(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)
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 = 0.20 \[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {-288+324 x-1458 x^2-216 x^3+8 x^4-9 x^5}{1230552 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {324 \log (x-\text {$\#$1})-2628 \log (x-\text {$\#$1}) \text {$\#$1}+324 \log (x-\text {$\#$1}) \text {$\#$1}^2-16 \log (x-\text {$\#$1}) \text {$\#$1}^3+9 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{7383312} \]
(-288 + 324*x - 1458*x^2 - 216*x^3 + 8*x^4 - 9*x^5)/(1230552*(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] - 2628*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) & ]/7383312
Time = 2.01 (sec) , antiderivative size = 826, normalized size of antiderivative = 0.97, 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.
| \(\displaystyle \int \frac {x^4}{\left (x^6+18 x^4+324 x^3+108 x^2+216\right )^2} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1586874322944 \int \left (-\frac {x+3\ 2^{2/3} \sqrt [3]{3}}{249875578388054016 \sqrt [3]{2} 3^{2/3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac {6 i 3^{5/6}-\left (\sqrt [3]{-2}+\sqrt [3]{2}\right ) x}{27763953154228224\ 6^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {\sqrt [3]{2} x+3^{5/6} \left (3 i-\sqrt {3}\right )}{124937789194027008\ 2^{2/3} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac {\sqrt [3]{-\frac {1}{3}}}{1542441841901568\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}}}{13881976577114112\ 2^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )^2}+\frac {1}{13881976577114112\ 2^{2/3} \sqrt [3]{3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1586874322944 \left (-\frac {\sqrt [3]{-\frac {1}{3}} \left (2 x+3 (-2)^{2/3} \sqrt [3]{3}\right )}{83291859462684672\ 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 {\left (1-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 )}{83291859462684672 \sqrt [6]{2} \sqrt [3]{3} \left (3 i+\sqrt {3}\right ) \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}-\frac {\sqrt [3]{-1} \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 )}{41645929731342336 \sqrt [6]{2} 3^{5/6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}+\frac {\sqrt [3]{-1} \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 )}{1156831381426176\ 2^{2/3} 3^{5/6} \left (1+\sqrt [3]{-1}\right )^4 \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 {\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 \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 )}{83291859462684672 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\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 )}{41645929731342336 \sqrt [6]{2} 3^{5/6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{55527906308456448 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac {\log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{249875578388054016 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right )}-\frac {\log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{499751156776108032 \sqrt [3]{2} 3^{2/3}}+\frac {\sqrt [3]{-\frac {1}{3}} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{9254651051409408 \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]{2} x+3 \sqrt [3]{3}}{83291859462684672 \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)*(3*(-3)^(1/3) - 2^(1/3)*x))/(9254651051409408 *(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)*(3*(-2)^(2/3)*3^(1/3) + 2*x))/(83291859462684672* 2^(2/3)*(4 + 3*(-2)^(1/3)*3^(2/3))*(6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2)) + ( 3*3^(1/3) + 2^(1/3)*x)/(83291859462684672*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)*ArcTan[(3*(-2)^(2/3)*3^(1/3 ) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(41645929731342336*2^(1/6)*3 ^(5/6)*(4 + 3*(-2)^(1/3)*3^(2/3))^(3/2)) - ((1 - I*Sqrt[3])*ArcTan[(3*(-2) ^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 + 3*(-2)^(1/3)*3^(2/3))]])/(83291859462684 672*2^(1/6)*3^(1/3)*(3*I + Sqrt[3])*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) - ((I/ 9254651051409408)*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)]) + ((-1)^(1/3)*ArcTan[(2^(1/6)*(3*(-3)^(1/3) - 2^(1/3)*x ))/Sqrt[3*(4 - 3*(-3)^(2/3)*2^(1/3))]])/(1156831381426176*2^(2/3)*3^(5/6)* (1 + (-1)^(1/3))^4*(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) )]]/(41645929731342336*2^(1/6)*3^(5/6)*(-4 + 3*2^(1/3)*3^(2/3))^(3/2)) + A rcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/3)*3^(2/3))]] /(83291859462684672*2^(1/6)*3^(5/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) - Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2]/(55527906308456448*2^(1/3)*3^(2/3)*(1 +...
3.2.55.3.1 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 | \(\frac {-\frac {1}{136728} x^{5}+\frac {1}{153819} x^{4}-\frac {1}{5697} x^{3}-\frac {1}{844} x^{2}+\frac {1}{3798} x -\frac {4}{17091}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-9 \textit {\_R}^{4}+16 \textit {\_R}^{3}-324 \textit {\_R}^{2}+2628 \textit {\_R} -324\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{7383312}\) | \(122\) |
| risch | \(\frac {-\frac {1}{136728} x^{5}+\frac {1}{153819} x^{4}-\frac {1}{5697} x^{3}-\frac {1}{844} x^{2}+\frac {1}{3798} x -\frac {4}{17091}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-9 \textit {\_R}^{4}+16 \textit {\_R}^{3}-324 \textit {\_R}^{2}+2628 \textit {\_R} -324\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{7383312}\) | \(122\) |
(-1/136728*x^5+1/153819*x^4-1/5697*x^3-1/844*x^2+1/3798*x-4/17091)/(x^6+18 *x^4+324*x^3+108*x^2+216)+1/7383312*sum((-9*_R^4+16*_R^3-324*_R^2+2628*_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. \[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.13 \[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\operatorname {RootSum} {\left (185583791958607219605834030755606257729536 t^{6} - 1309367357962223565522033377280 t^{4} + 4356336487052294744666112 t^{3} - 4052982845480387328 t^{2} + 303890718384 t - 880007, \left ( t \mapsto t \log {\left (\frac {39083462657955593476841044707333565976412952759280634691584 t^{5}}{49797855396139900267573395695} + \frac {8836979346223785538912817601414711102396804462575616 t^{4}}{49797855396139900267573395695} - \frac {264930581348308532588844249597134695706805067776 t^{3}}{49797855396139900267573395695} + \frac {886135333547363185201515109826158376250624 t^{2}}{49797855396139900267573395695} - \frac {682321479574909906511394635855601936 t}{49797855396139900267573395695} + x - \frac {21375560770846486224291519568}{49797855396139900267573395695} \right )} \right )\right )} + \frac {- 9 x^{5} + 8 x^{4} - 216 x^{3} - 1458 x^{2} + 324 x - 288}{1230552 x^{6} + 22149936 x^{4} + 398698848 x^{3} + 132899616 x^{2} + 265799232} \]
RootSum(185583791958607219605834030755606257729536*_t**6 - 130936735796222 3565522033377280*_t**4 + 4356336487052294744666112*_t**3 - 405298284548038 7328*_t**2 + 303890718384*_t - 880007, Lambda(_t, _t*log(39083462657955593 476841044707333565976412952759280634691584*_t**5/4979785539613990026757339 5695 + 8836979346223785538912817601414711102396804462575616*_t**4/49797855 396139900267573395695 - 264930581348308532588844249597134695706805067776*_ t**3/49797855396139900267573395695 + 8861353335473631852015151098261583762 50624*_t**2/49797855396139900267573395695 - 682321479574909906511394635855 601936*_t/49797855396139900267573395695 + x - 2137556077084648622429151956 8/49797855396139900267573395695))) + (-9*x**5 + 8*x**4 - 216*x**3 - 1458*x **2 + 324*x - 288)/(1230552*x**6 + 22149936*x**4 + 398698848*x**3 + 132899 616*x**2 + 265799232)
\[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{4}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
-1/1230552*(9*x^5 - 8*x^4 + 216*x^3 + 1458*x^2 - 324*x + 288)/(x^6 + 18*x^ 4 + 324*x^3 + 108*x^2 + 216) - 1/1230552*integrate((9*x^4 - 16*x^3 + 324*x ^2 - 2628*x + 324)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
\[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{4}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \]
Time = 9.10 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.46 \[ \int \frac {x^4}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Too large to display} \]
symsum(log((24389*root(z^6 - (60865*z^4)/8626729106138688 + (15496909*z^3) /660182046176944870474752 - (168169*z^2)/7700363386607884969217507328 + (3 971*z)/2425060040617647997585731147792384 - 880007/18558379195860721960583 4030755606257729536, z, k))/851770251396 + (288041*x)/804738905194918464 - (1090723*root(z^6 - (60865*z^4)/8626729106138688 + (15496909*z^3)/6601820 46176944870474752 - (168169*z^2)/7700363386607884969217507328 + (3971*z)/2 425060040617647997585731147792384 - 880007/1855837919586072196058340307556 06257729536, z, k)*x)/22997796787692 + (5850124*root(z^6 - (60865*z^4)/862 6729106138688 + (15496909*z^3)/660182046176944870474752 - (168169*z^2)/770 0363386607884969217507328 + (3971*z)/2425060040617647997585731147792384 - 880007/185583791958607219605834030755606257729536, z, k)^2*x)/3606201 - (6 4554687936*root(z^6 - (60865*z^4)/8626729106138688 + (15496909*z^3)/660182 046176944870474752 - (168169*z^2)/7700363386607884969217507328 + (3971*z)/ 2425060040617647997585731147792384 - 880007/185583791958607219605834030755 606257729536, z, k)^3*x)/44521 + (31535589897216*root(z^6 - (60865*z^4)/86 26729106138688 + (15496909*z^3)/660182046176944870474752 - (168169*z^2)/77 00363386607884969217507328 + (3971*z)/2425060040617647997585731147792384 - 880007/185583791958607219605834030755606257729536, z, k)^4*x)/211 - 69409 88288557056*root(z^6 - (60865*z^4)/8626729106138688 + (15496909*z^3)/66018 2046176944870474752 - (168169*z^2)/7700363386607884969217507328 + (3971...