\(\int \frac {x^8}{(216+108 x^2+324 x^3+18 x^4+x^6)^2} \, dx\) [23]

Optimal result

Integrand size = 26, antiderivative size = 1063 \[ \int \frac {x^8}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx =\text {Too large to display} \] Output:

-1/972*(-1)^(1/3)*3^(2/3)*(54+9*(-3)^(1/3)*2^(2/3)+(2-2^(2/3)*(6*(-6)^(2/3 
)+27*(-3)^(1/3)))*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*(-1)^(1/3)*3^(2/3)*(54-9*(-2)^(2/3)*3^( 
1/3)+(2+27*(-2)^(2/3)*3^(1/3)+12*(-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)+1/8748*(54- 
9*2^(2/3)*3^(1/3)+(2+2^(2/3)*(27*3^(1/3)-6*6^(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)-1/972*I*((-2)^(2/3)+6*3^(2 
/3))*arctan((3*(-2)^(2/3)*3^(1/3)+2*x)/(24+18*(-2)^(1/3)*3^(2/3))^(1/2))*2 
^(1/6)*3^(2/3)/(1+(-1)^(1/3))^5/(4+3*(-2)^(1/3)*3^(2/3))^(1/2)-1/972*(-1)^ 
(1/3)*(2+27*(-2)^(2/3)*3^(1/3)+12*(-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))*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)-1/486*(-1)^(1/3)*(6 
*(-6)^(2/3)+27*(-3)^(1/3)-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)-1/972*(9*3^(1/6)-I*(2^(2/3)-3*3^(2/3)))*arctan(2^( 
1/6)*(3*(-3)^(1/3)-2^(1/3)*x)/(12-9*(-3)^(2/3)*2^(1/3))^(1/2))*2^(1/6)*3^( 
2/3)/(1+(-1)^(1/3))^5/(4-3*(-3)^(2/3)*2^(1/3))^(1/2)-1/8748*(1+3*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 
))*2^(5/6)*3^(1/6)/(-4+3*2^(1/3)*3^(2/3))^(1/2)+1/486*(2^(1/3)+27*3^(1/3)- 
6*6^(2/3))*arctanh(2^(1/6)*(3*3^(1/3)+2^(1/3)*x)/(-12+9*2^(1/3)*3^(2/3)...
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.05 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.16 \[ \int \frac {x^8}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\frac {-7884+324 x-3990 x^2-11610 x^3-203 x^4-9 x^5}{34182 \left (216+108 x^2+324 x^3+18 x^4+x^6\right )}-\frac {\text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {324 \log (x-\text {$\#$1})-96 \log (x-\text {$\#$1}) \text {$\#$1}+324 \log (x-\text {$\#$1}) \text {$\#$1}^2+406 \log (x-\text {$\#$1}) \text {$\#$1}^3+9 \log (x-\text {$\#$1}) \text {$\#$1}^4}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ]}{205092} \] Input:

Integrate[x^8/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]
 

Output:

(-7884 + 324*x - 3990*x^2 - 11610*x^3 - 203*x^4 - 9*x^5)/(34182*(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] - 96*Log[x - #1]*#1 + 324*Log[x - #1]*#1^2 
 + 406*Log[x - #1]*#1^3 + 9*Log[x - #1]*#1^4)/(36*#1 + 162*#1^2 + 12*#1^3 
+ #1^5) & ]/205092
 

Rubi [A] (verified)

Time = 4.50 (sec) , antiderivative size = 1012, 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.

\(\displaystyle \int \frac {x^8}{\left (x^6+18 x^4+324 x^3+108 x^2+216\right )^2} \, dx\)

\(\Big \downarrow \) 2466

\(\displaystyle 1586874322944 \int \left (-\frac {i (27-x)}{771220920950784 \sqrt [3]{2} \sqrt [6]{3} \left (1+\sqrt [3]{-1}\right )^5 \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {27-x}{6940988288557056 \sqrt [3]{2} 3^{2/3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}+\frac {27 \left (2+(-1)^{2/3}\right )-\left (1+\sqrt [3]{-1}\right ) x}{771220920950784 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^5 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac {\sqrt [3]{-\frac {1}{3}} \left (-\left (\left (9+\sqrt [3]{-3} 2^{2/3}\right ) x\right )-3 (-3)^{2/3} \sqrt [3]{2}+1\right )}{42845606719488\ 2^{2/3} \left (1+\sqrt [3]{-1}\right )^4 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )^2}-\frac {\sqrt [3]{-\frac {1}{3}} \left (-\left (\left (9-(-2)^{2/3} \sqrt [3]{3}\right ) x\right )+3 \sqrt [3]{-2} 3^{2/3}+1\right )}{385610460475392\ 2^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )^2}+\frac {-\left (\left (9-2^{2/3} \sqrt [3]{3}\right ) x\right )-3 \sqrt [3]{2} 3^{2/3}+1}{385610460475392\ 2^{2/3} \sqrt [3]{3} \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 1586874322944 \left (-\frac {\sqrt [3]{-\frac {1}{3}} \left (\left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \sqrt [3]{-2} 3^{2/3}\right ) x+9 \left (6-(-2)^{2/3} \sqrt [3]{3}\right )\right )}{2313662762852352\ 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 {\sqrt [3]{-1} \left (2+27 (-2)^{2/3} \sqrt [3]{3}+12 \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 )}{2313662762852352 \sqrt [6]{2} 3^{5/6} \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )^{3/2}}-\frac {i \left ((-2)^{2/3}+6\ 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 )}{257073640316928\ 2^{5/6} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {\left (i 2^{2/3}-9 \sqrt [6]{3}-3 i 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 )}{257073640316928\ 2^{5/6} \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^5 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}-\frac {\sqrt [3]{-1} \left (6 (-6)^{2/3}+27 \sqrt [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 )}{128536820158464 \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 (\sqrt [3]{2}+27 \sqrt [3]{3}-6\ 6^{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 )}{1156831381426176 \sqrt {2} 3^{5/6} \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )^{3/2}}-\frac {\left (1+3 \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 )}{2313662762852352 \sqrt [6]{2} 3^{5/6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}-\frac {\log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{1542441841901568 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^4}+\frac {i \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{1542441841901568 \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 )}{13881976577114112 \sqrt [3]{2} 3^{2/3}}-\frac {\sqrt [3]{-\frac {1}{3}} \left (\left (2-3\ 2^{2/3} \left (2 (-6)^{2/3}+9 \sqrt [3]{-3}\right )\right ) x+9 \left (6+\sqrt [3]{-3} 2^{2/3}\right )\right )}{257073640316928\ 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 {\left (2+2^{2/3} \left (27 \sqrt [3]{3}-6\ 6^{2/3}\right )\right ) x+9 \left (6-2^{2/3} \sqrt [3]{3}\right )}{2313662762852352\ 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 )\)

Input:

Int[x^8/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)^2,x]
 

Output:

1586874322944*(-1/257073640316928*((-1/3)^(1/3)*(9*(6 + (-3)^(1/3)*2^(2/3) 
) + (2 - 3*2^(2/3)*(2*(-6)^(2/3) + 9*(-3)^(1/3)))*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)) - ( 
(-1/3)^(1/3)*(9*(6 - (-2)^(2/3)*3^(1/3)) + (2 + 27*(-2)^(2/3)*3^(1/3) + 12 
*(-2)^(1/3)*3^(2/3))*x))/(2313662762852352*2^(2/3)*(4 + 3*(-2)^(1/3)*3^(2/ 
3))*(6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2)) + (9*(6 - 2^(2/3)*3^(1/3)) + (2 + 
2^(2/3)*(27*3^(1/3) - 6*6^(2/3)))*x)/(2313662762852352*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)) - ((I/25707364031692 
8)*((-2)^(2/3) + 6*3^(2/3))*ArcTan[(3*(-2)^(2/3)*3^(1/3) + 2*x)/Sqrt[6*(4 
+ 3*(-2)^(1/3)*3^(2/3))]])/(2^(5/6)*3^(1/3)*(1 + (-1)^(1/3))^5*Sqrt[4 + 3* 
(-2)^(1/3)*3^(2/3)]) - ((-1)^(1/3)*(2 + 27*(-2)^(2/3)*3^(1/3) + 12*(-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))]])/(2313662762852352*2^(1/6)*3^(5/6)*(4 + 3*(-2)^(1/3)*3^(2/3))^( 
3/2)) - ((-1)^(1/3)*(6*(-6)^(2/3) + 27*(-3)^(1/3) - 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))]])/(12853 
6820158464*Sqrt[2]*3^(5/6)*(1 + (-1)^(1/3))^4*(4 - 3*(-3)^(2/3)*2^(1/3))^( 
3/2)) + ((I*2^(2/3) - 9*3^(1/6) - (3*I)*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))]])/(257073640316928*2 
^(5/6)*3^(1/3)*(1 + (-1)^(1/3))^5*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) - ((1 + 
3*2^(1/3)*3^(2/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2466

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]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.06 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.11

method	result	size
default	\(\frac {-\frac {1}{3798} x^{5}-\frac {203}{34182} x^{4}-\frac {215}{633} x^{3}-\frac {665}{5697} x^{2}+\frac {2}{211} x -\frac {146}{633}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-9 \textit {\_R}^{4}-406 \textit {\_R}^{3}-324 \textit {\_R}^{2}+96 \textit {\_R} -324\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{205092}\)	\(122\)
risch	\(\frac {-\frac {1}{3798} x^{5}-\frac {203}{34182} x^{4}-\frac {215}{633} x^{3}-\frac {665}{5697} x^{2}+\frac {2}{211} x -\frac {146}{633}}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-9 \textit {\_R}^{4}-406 \textit {\_R}^{3}-324 \textit {\_R}^{2}+96 \textit {\_R} -324\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{205092}\)	\(122\)

Input:

int(x^8/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x,method=_RETURNVERBOSE)
 

Output:

(-1/3798*x^5-203/34182*x^4-215/633*x^3-665/5697*x^2+2/211*x-146/633)/(x^6+ 
18*x^4+324*x^3+108*x^2+216)+1/205092*sum((-9*_R^4-406*_R^3-324*_R^2+96*_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))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^8}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Timed out} \] Input:

integrate(x^8/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.11 \[ \int \frac {x^8}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\operatorname {RootSum} {\left (85256017052964187415123360664576 t^{6} + 50105191533385434568704 t^{4} + 48885748051277486016 t^{3} + 865447782603408 t^{2} + 3220532460 t + 4513, \left ( t \mapsto t \log {\left (\frac {35492036204084174404119193135483487466590764032 t^{5}}{356900697070792948475845} - \frac {19474160067218837086826809631017022308224 t^{4}}{71380139414158589695169} + \frac {20779963076545132233894582764903396544 t^{3}}{356900697070792948475845} + \frac {20265219154367004972162198012037344 t^{2}}{356900697070792948475845} + \frac {275192468949210532049075145372 t}{356900697070792948475845} + x + \frac {1290285191292177289622012}{1070702091212378845427535} \right )} \right )\right )} + \frac {- 9 x^{5} - 203 x^{4} - 11610 x^{3} - 3990 x^{2} + 324 x - 7884}{34182 x^{6} + 615276 x^{4} + 11074968 x^{3} + 3691656 x^{2} + 7383312} \] Input:

integrate(x**8/(x**6+18*x**4+324*x**3+108*x**2+216)**2,x)
 

Output:

RootSum(85256017052964187415123360664576*_t**6 + 50105191533385434568704*_ 
t**4 + 48885748051277486016*_t**3 + 865447782603408*_t**2 + 3220532460*_t 
+ 4513, Lambda(_t, _t*log(35492036204084174404119193135483487466590764032* 
_t**5/356900697070792948475845 - 19474160067218837086826809631017022308224 
*_t**4/71380139414158589695169 + 20779963076545132233894582764903396544*_t 
**3/356900697070792948475845 + 20265219154367004972162198012037344*_t**2/3 
56900697070792948475845 + 275192468949210532049075145372*_t/35690069707079 
2948475845 + x + 1290285191292177289622012/1070702091212378845427535))) + 
(-9*x**5 - 203*x**4 - 11610*x**3 - 3990*x**2 + 324*x - 7884)/(34182*x**6 + 
 615276*x**4 + 11074968*x**3 + 3691656*x**2 + 7383312)
 

Maxima [F]

\[ \int \frac {x^8}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{8}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \] Input:

integrate(x^8/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="maxima")
 

Output:

-1/34182*(9*x^5 + 203*x^4 + 11610*x^3 + 3990*x^2 - 324*x + 7884)/(x^6 + 18 
*x^4 + 324*x^3 + 108*x^2 + 216) - 1/34182*integrate((9*x^4 + 406*x^3 + 324 
*x^2 - 96*x + 324)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
 

Giac [F]

\[ \int \frac {x^8}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\int { \frac {x^{8}}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )}^{2}} \,d x } \] Input:

integrate(x^8/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x, algorithm="giac")
 

Output:

integrate(x^8/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)^2, x)
                                                                                    
                                                                                    
 

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.37 \[ \int \frac {x^8}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {Too large to display} \] Input:

int(x^8/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216)^2,x)
 

Output:

symsum(log((239491904*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14 
149992416343982992 + (5171*z^2)/509399726988383387712 + (505*z)/1336868643 
5083133627113728 + 4513/85256017052964187415123360664576, z, k)*x)/8763068 
43 - (275536*x)/638827688547 - (3848128*root(z^6 + (326*z^4)/554702231619 
+ (8113597*z^3)/14149992416343982992 + (5171*z^2)/509399726988383387712 + 
(505*z)/13368686435083133627113728 + 4513/85256017052964187415123360664576 
, z, k))/3606201 - (152363520*root(z^6 + (326*z^4)/554702231619 + (8113597 
*z^3)/14149992416343982992 + (5171*z^2)/509399726988383387712 + (505*z)/13 
368686435083133627113728 + 4513/85256017052964187415123360664576, z, k)^2* 
x)/44521 - (698075283456*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3) 
/14149992416343982992 + (5171*z^2)/509399726988383387712 + (505*z)/1336868 
6435083133627113728 + 4513/85256017052964187415123360664576, z, k)^3*x)/44 
521 + (130789789876224*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/1 
4149992416343982992 + (5171*z^2)/509399726988383387712 + (505*z)/133686864 
35083133627113728 + 4513/85256017052964187415123360664576, z, k)^4*x)/211 
- 6940988288557056*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/14149 
992416343982992 + (5171*z^2)/509399726988383387712 + (505*z)/1336868643508 
3133627113728 + 4513/85256017052964187415123360664576, z, k)^5*x - (426422 
0928*root(z^6 + (326*z^4)/554702231619 + (8113597*z^3)/1414999241634398299 
2 + (5171*z^2)/509399726988383387712 + (505*z)/133686864350831336271137...
 

Reduce [F]

\[ \int \frac {x^8}{\left (216+108 x^2+324 x^3+18 x^4+x^6\right )^2} \, dx=\text {too large to display} \] Input:

int(x^8/(x^6+18*x^4+324*x^3+108*x^2+216)^2,x)
 

Output:

( - 134946*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 - 2429028*int(x**10/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11 
664*x**7 + 109296*x**6 + 69984*x**5 + 19440*x**4 + 139968*x**3 + 46656*x** 
2 + 46656),x)*x**4 - 43722504*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**3 - 14574168*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 + 139 
968*x**3 + 46656*x**2 + 46656),x)*x**2 - 29148336*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) + 269892*int(x**9/(x**12 + 36* 
x**10 + 648*x**9 + 540*x**8 + 11664*x**7 + 109296*x**6 + 69984*x**5 + 1944 
0*x**4 + 139968*x**3 + 46656*x**2 + 46656),x)*x**6 + 4858056*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**4 + 87445008*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 + 2914 
8336*int(x**9/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11664*x**7 + 10929 
6*x**6 + 69984*x**5 + 19440*x**4 + 139968*x**3 + 46656*x**2 + 46656),x)*x* 
*2 + 58296672*int(x**9/(x**12 + 36*x**10 + 648*x**9 + 540*x**8 + 11664*...