Integrand size = 30, antiderivative size = 610 \[ \int \frac {1}{(2+3 x) \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=-\frac {(-1)^{2/3} \left (166-27 (-2)^{2/3} \sqrt [3]{3}+6 \sqrt [3]{-2} 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{648\ 2^{5/6} \sqrt [6]{3} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (29-9 (-2)^{2/3} \sqrt [3]{3}\right ) \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {(-1)^{2/3} \left (3 (-6)^{2/3}-27 \sqrt [3]{-3}-83 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{648 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}} \left (29+9 \sqrt [3]{-3} 2^{2/3}\right )}+\frac {\left (83 \sqrt [3]{2}-27 \sqrt [3]{3}-3\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{648 \sqrt [6]{6} \left (1-\sqrt [3]{-1}\right ) \left (1+\sqrt [3]{-1}\right )^2 \left (29-9\ 2^{2/3} \sqrt [3]{3}\right ) \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {243 \log (2+3 x)}{125128}-\frac {\left ((-6)^{2/3}+18 \sqrt [3]{-3}+2 \sqrt [3]{2}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{864 \sqrt [3]{3} \left (1+\sqrt [3]{-1}\right )^2 \left (29+9 \sqrt [3]{-3} 2^{2/3}\right )}+\frac {(-1)^{2/3} \left (9 \sqrt [3]{-6}+(-2)^{2/3}+3^{2/3}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{6} \left (58+9\ 2^{2/3} \sqrt [3]{3}-9 i 2^{2/3} 3^{5/6}\right )}-\frac {\left (54-6^{2/3} \left (2^{2/3}+3^{2/3}\right )\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{7776 \left (29-9\ 2^{2/3} \sqrt [3]{3}\right )} \] Output:
-1/3888*(-1)^(2/3)*(166-27*(-2)^(2/3)*3^(1/3)+6*(-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^(1/6)*3^(5 /6)/(1-(-1)^(1/3))/(1+(-1)^(1/3))^2/(29-9*(-2)^(2/3)*3^(1/3))/(4+3*(-2)^(1 /3)*3^(2/3))^(1/2)+1/3888*(-1)^(2/3)*(3*(-6)^(2/3)-27*(-3)^(1/3)-83*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) )*6^(5/6)/(1+(-1)^(1/3))^2/(4-3*(-3)^(2/3)*2^(1/3))^(1/2)/(29+9*(-3)^(1/3) *2^(2/3))+1/3888*(83*2^(1/3)-27*3^(1/3)-3*6^(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^(5/6)/(1-(-1)^(1/3))/(1+(- 1)^(1/3))^2/(29-9*2^(2/3)*3^(1/3))/(-4+3*2^(1/3)*3^(2/3))^(1/2)+243/125128 *ln(2+3*x)-1/2592*((-6)^(2/3)+18*(-3)^(1/3)+2*2^(1/3))*ln(6-3*(-3)^(1/3)*2 ^(2/3)*x+x^2)*3^(2/3)/(1+(-1)^(1/3))^2/(29+9*(-3)^(1/3)*2^(2/3))+1/3888*(- 1)^(2/3)*(9*(-6)^(1/3)+(-2)^(2/3)+3^(2/3))*ln(6+3*(-2)^(2/3)*3^(1/3)*x+x^2 )*6^(2/3)/(58+9*2^(2/3)*3^(1/3)-9*I*2^(2/3)*3^(5/6))-(54-6^(2/3)*(2^(2/3)+ 3^(2/3)))*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)/(225504-69984*2^(2/3)*3^(1/3))
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.04 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.28 \[ \int \frac {1}{(2+3 x) \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\frac {243 \log (2+3 x)}{125128}-\frac {81 \text {RootSum}\left [125128+64608 \text {$\#$1}-39612 \text {$\#$1}^2+7292 \text {$\#$1}^3+222 \text {$\#$1}^4-12 \text {$\#$1}^5+\text {$\#$1}^6\&,\frac {64608 \log (2+3 x-\text {$\#$1})-39612 \log (2+3 x-\text {$\#$1}) \text {$\#$1}+7292 \log (2+3 x-\text {$\#$1}) \text {$\#$1}^2+222 \log (2+3 x-\text {$\#$1}) \text {$\#$1}^3-12 \log (2+3 x-\text {$\#$1}) \text {$\#$1}^4+\log (2+3 x-\text {$\#$1}) \text {$\#$1}^5}{10768-13204 \text {$\#$1}+3646 \text {$\#$1}^2+148 \text {$\#$1}^3-10 \text {$\#$1}^4+\text {$\#$1}^5}\&\right ]}{250256} \] Input:
Integrate[1/((2 + 3*x)*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)),x]
Output:
(243*Log[2 + 3*x])/125128 - (81*RootSum[125128 + 64608*#1 - 39612*#1^2 + 7 292*#1^3 + 222*#1^4 - 12*#1^5 + #1^6 & , (64608*Log[2 + 3*x - #1] - 39612* Log[2 + 3*x - #1]*#1 + 7292*Log[2 + 3*x - #1]*#1^2 + 222*Log[2 + 3*x - #1] *#1^3 - 12*Log[2 + 3*x - #1]*#1^4 + Log[2 + 3*x - #1]*#1^5)/(10768 - 13204 *#1 + 3646*#1^2 + 148*#1^3 - 10*#1^4 + #1^5) & ])/250256
Time = 2.97 (sec) , antiderivative size = 592, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, 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.
| \(\displaystyle \int \frac {1}{(3 x+2) \left (x^6+18 x^4+324 x^3+108 x^2+216\right )} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1259712 \int \left (\frac {(-1)^{2/3} \left (2 \left (82-3 (-3)^{2/3} \sqrt [3]{2}+9 \sqrt [3]{-3} 2^{2/3}\right )-\left (3-9 (-3)^{2/3} \sqrt [3]{2}-\sqrt [3]{-3} 2^{2/3}\right ) x\right )}{272097792 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (29+9 \sqrt [3]{-3} 2^{2/3}\right ) \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}+\frac {1}{216221184 (3 x+2)}-\frac {(-1)^{2/3} \left (2 \left (82-9 (-2)^{2/3} \sqrt [3]{3}+3 \sqrt [3]{-2} 3^{2/3}\right )-\left (3+\sqrt [3]{3} \left (9 \sqrt [3]{-6}+(-2)^{2/3}\right )\right ) x\right )}{408146688 \sqrt [3]{2} 3^{2/3} \left (58+9\ 2^{2/3} \sqrt [3]{3}-9 i 2^{2/3} 3^{5/6}\right ) \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}-\frac {2 \left (82-9\ 2^{2/3} \sqrt [3]{3}-3 \sqrt [3]{2} 3^{2/3}\right )-\left (3+\sqrt [3]{3} \left (2^{2/3}-9 \sqrt [3]{6}\right )\right ) x}{816293376 \sqrt [3]{2} 3^{2/3} \left (29-9\ 2^{2/3} \sqrt [3]{3}\right ) \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1259712 \left (\frac {\sqrt [6]{-\frac {1}{3}} \left (166-27 (-2)^{2/3} \sqrt [3]{3}+6 \sqrt [3]{-2} 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1224440064\ 2^{5/6} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}} \left (58 i+9 i 2^{2/3} \sqrt [3]{3}+9\ 2^{2/3} 3^{5/6}\right )}+\frac {(-1)^{2/3} \left (3 (-6)^{2/3}-27 \sqrt [3]{-3}-83 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{816293376 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}} \left (29+9 \sqrt [3]{-3} 2^{2/3}\right )}+\frac {\left (83 \sqrt [3]{2}-27 \sqrt [3]{3}-3\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{2448880128 \sqrt [6]{6} \left (29-9\ 2^{2/3} \sqrt [3]{3}\right ) \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}-\frac {\left (54 \sqrt [3]{-1}+3 (-2)^{2/3} \sqrt [3]{3}+2 \sqrt [3]{2} 3^{2/3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{3265173504 \left (1+\sqrt [3]{-1}\right )^2 \left (29+9 \sqrt [3]{-3} 2^{2/3}\right )}+\frac {(-1)^{2/3} \left (9 \sqrt [3]{-6}+(-2)^{2/3}+3^{2/3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{816293376 \sqrt [3]{6} \left (58+9\ 2^{2/3} \sqrt [3]{3}-9 i 2^{2/3} 3^{5/6}\right )}-\frac {\left (54-6^{2/3} \left (2^{2/3}+3^{2/3}\right )\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{9795520512 \left (29-9\ 2^{2/3} \sqrt [3]{3}\right )}+\frac {\log (3 x+2)}{648663552}\right )\) |
Input:
Int[1/((2 + 3*x)*(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6)),x]
Output:
1259712*(((-1/3)^(1/6)*(166 - 27*(-2)^(2/3)*3^(1/3) + 6*(-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))]]) /(1224440064*2^(5/6)*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]*(58*I + (9*I)*2^(2/3)* 3^(1/3) + 9*2^(2/3)*3^(5/6))) + ((-1)^(2/3)*(3*(-6)^(2/3) - 27*(-3)^(1/3) - 83*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))]])/(816293376*6^(1/6)*(1 + (-1)^(1/3))^2*Sqrt[4 - 3*(-3) ^(2/3)*2^(1/3)]*(29 + 9*(-3)^(1/3)*2^(2/3))) + ((83*2^(1/3) - 27*3^(1/3) - 3*6^(2/3))*ArcTanh[(2^(1/6)*(3*3^(1/3) + 2^(1/3)*x))/Sqrt[3*(-4 + 3*2^(1/ 3)*3^(2/3))]])/(2448880128*6^(1/6)*(29 - 9*2^(2/3)*3^(1/3))*Sqrt[-4 + 3*2^ (1/3)*3^(2/3)]) + Log[2 + 3*x]/648663552 - ((54*(-1)^(1/3) + 3*(-2)^(2/3)* 3^(1/3) + 2*2^(1/3)*3^(2/3))*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(32651 73504*(1 + (-1)^(1/3))^2*(29 + 9*(-3)^(1/3)*2^(2/3))) + ((-1)^(2/3)*(9*(-6 )^(1/3) + (-2)^(2/3) + 3^(2/3))*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(81 6293376*6^(1/3)*(58 + 9*2^(2/3)*3^(1/3) - (9*I)*2^(2/3)*3^(5/6))) - ((54 - 6^(2/3)*(2^(2/3) + 3^(2/3)))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/(9795520 512*(29 - 9*2^(2/3)*3^(1/3))))
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 = 0.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.13
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (79206024 \textit {\_Z}^{6}+299024136 \textit {\_Z}^{5}+290189628 \textit {\_Z}^{4}+93944556 \textit {\_Z}^{3}-764154 \textit {\_Z}^{2}+1530 \textit {\_Z} -1\right )}{\sum }\textit {\_R} \ln \left (-133339913862864047026951648912056 \textit {\_R}^{5}-503455034796500794783556665488168 \textit {\_R}^{4}-488751107538106925077309379679144 \textit {\_R}^{3}-158373383880682539034481169501120 \textit {\_R}^{2}+1215540060612927053513490370296 \textit {\_R} +58850476781569891989955805 x -1531118504941184606171339196\right )\right )}{1944}+\frac {243 \ln \left (2+3 x \right )}{125128}\) | \(77\) |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (-243 \textit {\_R}^{5}+162 \textit {\_R}^{4}-4482 \textit {\_R}^{3}-75744 \textit {\_R}^{2}+24252 \textit {\_R} -16168\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{750768}+\frac {243 \ln \left (2+3 x \right )}{125128}\) | \(87\) |
Input:
int(1/(2+3*x)/(x^6+18*x^4+324*x^3+108*x^2+216),x,method=_RETURNVERBOSE)
Output:
1/1944*sum(_R*ln(-133339913862864047026951648912056*_R^5-50345503479650079 4783556665488168*_R^4-488751107538106925077309379679144*_R^3-1583733838806 82539034481169501120*_R^2+1215540060612927053513490370296*_R+5885047678156 9891989955805*x-1531118504941184606171339196),_R=RootOf(79206024*_Z^6+2990 24136*_Z^5+290189628*_Z^4+93944556*_Z^3-764154*_Z^2+1530*_Z-1))+243/125128 *ln(2+3*x)
Timed out. \[ \int \frac {1}{(2+3 x) \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(2+3*x)/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="fricas" )
Output:
Timed out
Time = 0.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.14 \[ \int \frac {1}{(2+3 x) \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\frac {243 \log {\left (x + \frac {2}{3} \right )}}{125128} + \operatorname {RootSum} {\left (4274996628704706944191954944 t^{6} + 8302092103887569428414464 t^{5} + 4144450225120841023488 t^{4} + 690176868966549504 t^{3} - 2887841890944 t^{2} + 2974320 t - 1, \left ( t \mapsto t \log {\left (\frac {186549983792253452567960080749168217215089045324444510832167064251139373466237534208 t^{6}}{5758416557677247627715162336355749658352447806959442529575} + \frac {43725890271624606138420190534428681871298998400865601422889352424228841586688 t^{5}}{5758416557677247627715162336355749658352447806959442529575} - \frac {522702975661454177912290454139420738467736654309175195594878675614610395234304 t^{4}}{5758416557677247627715162336355749658352447806959442529575} - \frac {321224079878547372471171470938499880967240473547842617172308647169720205312 t^{3}}{5758416557677247627715162336355749658352447806959442529575} - \frac {58689624023686849959854698156402758358057666085552221408970250486091008 t^{2}}{5758416557677247627715162336355749658352447806959442529575} + \frac {9253847552744655196468299940585461581206375367903425714731382944 t}{230336662307089905108606493454229986334097912278377701183} + x - \frac {449582741794983102129276747942184631844470423549131141927666}{17275249673031742883145487009067248975057343420878327588725} \right )} \right )\right )} \] Input:
integrate(1/(2+3*x)/(x**6+18*x**4+324*x**3+108*x**2+216),x)
Output:
243*log(x + 2/3)/125128 + RootSum(4274996628704706944191954944*_t**6 + 830 2092103887569428414464*_t**5 + 4144450225120841023488*_t**4 + 690176868966 549504*_t**3 - 2887841890944*_t**2 + 2974320*_t - 1, Lambda(_t, _t*log(186 54998379225345256796008074916821721508904532444451083216706425113937346623 7534208*_t**6/5758416557677247627715162336355749658352447806959442529575 + 4372589027162460613842019053442868187129899840086560142288935242422884158 6688*_t**5/5758416557677247627715162336355749658352447806959442529575 - 52 27029756614541779122904541394207384677366543091751955948786756146103952343 04*_t**4/5758416557677247627715162336355749658352447806959442529575 - 3212 24079878547372471171470938499880967240473547842617172308647169720205312*_t **3/5758416557677247627715162336355749658352447806959442529575 - 586896240 23686849959854698156402758358057666085552221408970250486091008*_t**2/57584 16557677247627715162336355749658352447806959442529575 + 925384755274465519 6468299940585461581206375367903425714731382944*_t/230336662307089905108606 493454229986334097912278377701183 + x - 4495827417949831021292767479421846 31844470423549131141927666/17275249673031742883145487009067248975057343420 878327588725)))
\[ \int \frac {1}{(2+3 x) \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} {\left (3 \, x + 2\right )}} \,d x } \] Input:
integrate(1/(2+3*x)/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima" )
Output:
-1/125128*integrate((243*x^5 - 162*x^4 + 4482*x^3 + 75744*x^2 - 24252*x + 16168)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x) + 243/125128*log(3*x + 2)
\[ \int \frac {1}{(2+3 x) \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int { \frac {1}{{\left (x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216\right )} {\left (3 \, x + 2\right )}} \,d x } \] Input:
integrate(1/(2+3*x)/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="giac")
Output:
integrate(1/((x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216)*(3*x + 2)), x)
Time = 22.15 (sec) , antiderivative size = 467, normalized size of antiderivative = 0.77 \[ \int \frac {1}{(2+3 x) \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\text {Too large to display} \] Input:
int(1/((3*x + 2)*(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216)),x)
Output:
(243*log(x + 2/3))/125128 + symsum(log((2*root(z^6 + (243*z^5)/125128 + (2 98549*z^4)/307953021312 + (869857*z^3)/5387946060874752 - (4717*z^2)/69827 78094893678592 + (85*z)/122170685548259800645632 - 1/427499662870470694419 1954944, z, k))/729 + (7*root(z^6 + (243*z^5)/125128 + (298549*z^4)/307953 021312 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 + (85*z)/122170685548259800645632 - 1/4274996628704706944191954944, z, k)*x) /243 - (1445769776*root(z^6 + (243*z^5)/125128 + (298549*z^4)/307953021312 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 + (85*z) /122170685548259800645632 - 1/4274996628704706944191954944, z, k)^2*x)/196 83 + (1605207476480*root(z^6 + (243*z^5)/125128 + (298549*z^4)/30795302131 2 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 + (85*z )/122170685548259800645632 - 1/4274996628704706944191954944, z, k)^3*x)/27 - 11385414082473984*root(z^6 + (243*z^5)/125128 + (298549*z^4)/3079530213 12 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 + (85* z)/122170685548259800645632 - 1/4274996628704706944191954944, z, k)^4*x - 46778486686192041984*root(z^6 + (243*z^5)/125128 + (298549*z^4)/3079530213 12 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 + (85* z)/122170685548259800645632 - 1/4274996628704706944191954944, z, k)^5*x - 50765351266770267930624*root(z^6 + (243*z^5)/125128 + (298549*z^4)/3079530 21312 + (869857*z^3)/5387946060874752 - (4717*z^2)/6982778094893678592 ...
\[ \int \frac {1}{(2+3 x) \left (216+108 x^2+324 x^3+18 x^4+x^6\right )} \, dx=\int \frac {1}{3 x^{7}+2 x^{6}+54 x^{5}+1008 x^{4}+972 x^{3}+216 x^{2}+648 x +432}d x \] Input:
int(1/(2+3*x)/(x^6+18*x^4+324*x^3+108*x^2+216),x)
Output:
int(1/(3*x**7 + 2*x**6 + 54*x**5 + 1008*x**4 + 972*x**3 + 216*x**2 + 648*x + 432),x)