Integrand size = 28, antiderivative size = 441 \[ \int \frac {2+3 x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {\left (9 \sqrt [3]{-6}-2 (-2)^{2/3}+6\ 3^{2/3}\right ) \arctan \left (\frac {3 (-2)^{2/3} \sqrt [3]{3}+2 x}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{972 \sqrt [6]{3} \sqrt {8+9 i \sqrt [3]{2} \sqrt [6]{3}+3 \sqrt [3]{2} 3^{2/3}}}+\frac {(-1)^{2/3} \left (3 (-6)^{2/3}+9 \sqrt [3]{-3}-2 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{324 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\left (2 \sqrt [3]{2}+9 \sqrt [3]{3}-3\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{3}+\sqrt [3]{2} x\right )}{\sqrt {3 \left (-4+3 \sqrt [3]{2} 3^{2/3}\right )}}\right )}{972 \sqrt [6]{6} \sqrt {-4+3 \sqrt [3]{2} 3^{2/3}}}+\frac {(-1)^{2/3} \left (3+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (6-3 \sqrt [3]{-3} 2^{2/3} x+x^2\right )}{216 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}-\frac {\left ((-6)^{2/3}+2 \sqrt [3]{-2}\right ) \log \left (6+3 (-2)^{2/3} \sqrt [3]{3} x+x^2\right )}{1296 \sqrt [3]{3}}+\frac {\left (2^{2/3}-3^{2/3}\right ) \log \left (6+3\ 2^{2/3} \sqrt [3]{3} x+x^2\right )}{648 \sqrt [3]{6}} \] Output:
1/2916*(9*(-6)^(1/3)-2*(-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))*3^(5/6)/(8+9*I*2^(1/3)*3^(1/6)+3*2^ (1/3)*3^(2/3))^(1/2)+1/1944*(-1)^(2/3)*(3*(-6)^(2/3)+9*(-3)^(1/3)-2*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)+1/5832*(2*2^(1/3 )+9*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)/(-4+3*2^(1/3)*3^(2/3))^(1/2)+1/1296*(-1)^(2/3)*( 3+(-3)^(1/3)*2^(2/3))*ln(6-3*(-3)^(1/3)*2^(2/3)*x+x^2)*2^(2/3)*3^(1/3)/(1+ (-1)^(1/3))^2-1/3888*((-6)^(2/3)+2*(-2)^(1/3))*ln(6+3*(-2)^(2/3)*3^(1/3)*x +x^2)*3^(2/3)+1/3888*(2^(2/3)-3^(2/3))*ln(6+3*2^(2/3)*3^(1/3)*x+x^2)*6^(2/ 3)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.17 \[ \int \frac {2+3 x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\frac {1}{6} \text {RootSum}\left [216+108 \text {$\#$1}^2+324 \text {$\#$1}^3+18 \text {$\#$1}^4+\text {$\#$1}^6\&,\frac {2 \log (x-\text {$\#$1})+3 \log (x-\text {$\#$1}) \text {$\#$1}}{36 \text {$\#$1}+162 \text {$\#$1}^2+12 \text {$\#$1}^3+\text {$\#$1}^5}\&\right ] \] Input:
Integrate[(2 + 3*x)/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
Output:
RootSum[216 + 108*#1^2 + 324*#1^3 + 18*#1^4 + #1^6 & , (2*Log[x - #1] + 3* Log[x - #1]*#1)/(36*#1 + 162*#1^2 + 12*#1^3 + #1^5) & ]/6
Time = 2.14 (sec) , antiderivative size = 448, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, 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.
| \(\displaystyle \int \frac {3 x+2}{x^6+18 x^4+324 x^3+108 x^2+216} \, dx\) |
| \(\Big \downarrow \) 2466 |
| \(\displaystyle 1259712 \int \left (\frac {(-1)^{2/3} \left (\left (3+\sqrt [3]{-3} 2^{2/3}\right ) x-9 \sqrt [3]{-3} 2^{2/3}-6 (-3)^{2/3} \sqrt [3]{2}+2\right )}{136048896 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2 \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}-\frac {(-1)^{2/3} \left (\left (3-(-2)^{2/3} \sqrt [3]{3}\right ) x+6 \sqrt [3]{-2} 3^{2/3}+9 (-2)^{2/3} \sqrt [3]{3}+2\right )}{408146688 \sqrt [3]{2} 3^{2/3} \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}+\frac {6^{2/3} \left (2^{2/3}-3^{2/3}\right ) x+2 \left (18-2^{2/3} \sqrt [3]{3}-9 \sqrt [3]{2} 3^{2/3}\right )}{2448880128 \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 1259712 \left (-\frac {(-1)^{2/3} \left (4+9 (-2)^{2/3} \sqrt [3]{3}+6 \sqrt [3]{-2} 3^{2/3}\right ) \arctan \left (\frac {2 x+3 (-2)^{2/3} \sqrt [3]{3}}{\sqrt {6 \left (4+3 \sqrt [3]{-2} 3^{2/3}\right )}}\right )}{1224440064\ 2^{5/6} \sqrt [6]{3} \sqrt {4+3 \sqrt [3]{-2} 3^{2/3}}}+\frac {(-1)^{2/3} \left (3 (-6)^{2/3}+9 \sqrt [3]{-3}-2 \sqrt [3]{2}\right ) \arctan \left (\frac {\sqrt [6]{2} \left (3 \sqrt [3]{-3}-\sqrt [3]{2} x\right )}{\sqrt {3 \left (4-3 (-3)^{2/3} \sqrt [3]{2}\right )}}\right )}{408146688 \sqrt [6]{6} \left (1+\sqrt [3]{-1}\right )^2 \sqrt {4-3 (-3)^{2/3} \sqrt [3]{2}}}+\frac {\left (2 \sqrt [3]{2}+9 \sqrt [3]{3}-3\ 6^{2/3}\right ) \text {arctanh}\left (\frac {\sqrt [6]{2} \left (\sqrt [3]{2} x+3 \sqrt [3]{3}\right )}{\sqrt {3 \left (3 \sqrt [3]{2} 3^{2/3}-4\right )}}\right )}{1224440064 \sqrt [6]{6} \sqrt {3 \sqrt [3]{2} 3^{2/3}-4}}+\frac {(-1)^{2/3} \left (3+\sqrt [3]{-3} 2^{2/3}\right ) \log \left (x^2-3 \sqrt [3]{-3} 2^{2/3} x+6\right )}{272097792 \sqrt [3]{2} 3^{2/3} \left (1+\sqrt [3]{-1}\right )^2}+\frac {(-1)^{2/3} \left ((-2)^{2/3}-3^{2/3}\right ) \log \left (x^2+3 (-2)^{2/3} \sqrt [3]{3} x+6\right )}{816293376 \sqrt [3]{6}}+\frac {\left (2^{2/3}-3^{2/3}\right ) \log \left (x^2+3\ 2^{2/3} \sqrt [3]{3} x+6\right )}{816293376 \sqrt [3]{6}}\right )\) |
Input:
Int[(2 + 3*x)/(216 + 108*x^2 + 324*x^3 + 18*x^4 + x^6),x]
Output:
1259712*(-1/1224440064*((-1)^(2/3)*(4 + 9*(-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))]])/(2^(5/6)*3^(1/6)*Sqrt[4 + 3*(-2)^(1/3)*3^(2/3)]) + ((-1)^(2/3)*( 3*(-6)^(2/3) + 9*(-3)^(1/3) - 2*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))]])/(408146688*6^(1/6)*(1 + (- 1)^(1/3))^2*Sqrt[4 - 3*(-3)^(2/3)*2^(1/3)]) + ((2*2^(1/3) + 9*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))]])/(1224440064*6^(1/6)*Sqrt[-4 + 3*2^(1/3)*3^(2/3)]) + ((-1)^(2/3 )*(3 + (-3)^(1/3)*2^(2/3))*Log[6 - 3*(-3)^(1/3)*2^(2/3)*x + x^2])/(2720977 92*2^(1/3)*3^(2/3)*(1 + (-1)^(1/3))^2) + ((-1)^(2/3)*((-2)^(2/3) - 3^(2/3) )*Log[6 + 3*(-2)^(2/3)*3^(1/3)*x + x^2])/(816293376*6^(1/3)) + ((2^(2/3) - 3^(2/3))*Log[6 + 3*2^(2/3)*3^(1/3)*x + x^2])/(816293376*6^(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 = 0.04 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.13
| method | result | size |
| default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (2+3 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(58\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}+18 \textit {\_Z}^{4}+324 \textit {\_Z}^{3}+108 \textit {\_Z}^{2}+216\right )}{\sum }\frac {\left (2+3 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}+12 \textit {\_R}^{3}+162 \textit {\_R}^{2}+36 \textit {\_R}}\right )}{6}\) | \(58\) |
Input:
int((2+3*x)/(x^6+18*x^4+324*x^3+108*x^2+216),x,method=_RETURNVERBOSE)
Output:
1/6*sum((2+3*_R)/(_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. \[ \int \frac {2+3 x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\text {Timed out} \] Input:
integrate((2+3*x)/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="fricas")
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.15 \[ \int \frac {2+3 x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\operatorname {RootSum} {\left (4270623510230231187456 t^{6} + 101511953720146944 t^{4} + 109196789347584 t^{3} - 81578319264 t^{2} - 1609632 t - 15641, \left ( t \mapsto t \log {\left (\frac {6219534773825866488077542999646208 t^{5}}{8049794570608956253} + \frac {955588046684567775465000499200 t^{4}}{8049794570608956253} + \frac {13409641654345971318505496832 t^{3}}{731799506418996023} + \frac {182573053374377754517021632 t^{2}}{8049794570608956253} - \frac {9200303300062511923176 t}{731799506418996023} + x - \frac {11781507074505066000}{8049794570608956253} \right )} \right )\right )} \] Input:
integrate((2+3*x)/(x**6+18*x**4+324*x**3+108*x**2+216),x)
Output:
RootSum(4270623510230231187456*_t**6 + 101511953720146944*_t**4 + 10919678 9347584*_t**3 - 81578319264*_t**2 - 1609632*_t - 15641, Lambda(_t, _t*log( 6219534773825866488077542999646208*_t**5/8049794570608956253 + 95558804668 4567775465000499200*_t**4/8049794570608956253 + 13409641654345971318505496 832*_t**3/731799506418996023 + 182573053374377754517021632*_t**2/804979457 0608956253 - 9200303300062511923176*_t/731799506418996023 + x - 1178150707 4505066000/8049794570608956253)))
\[ \int \frac {2+3 x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {3 \, x + 2}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate((2+3*x)/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="maxima")
Output:
integrate((3*x + 2)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
\[ \int \frac {2+3 x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=\int { \frac {3 \, x + 2}{x^{6} + 18 \, x^{4} + 324 \, x^{3} + 108 \, x^{2} + 216} \,d x } \] Input:
integrate((2+3*x)/(x^6+18*x^4+324*x^3+108*x^2+216),x, algorithm="giac")
Output:
integrate((3*x + 2)/(x^6 + 18*x^4 + 324*x^3 + 108*x^2 + 216), x)
Time = 22.33 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.77 \[ \int \frac {2+3 x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx =\text {Too large to display} \] Input:
int((3*x + 2)/(108*x^2 + 324*x^3 + 18*x^4 + x^6 + 216),x)
Output:
symsum(log(81*x - 26640*root(z^6 + (13*z^4)/546912 + (367*z^3)/14353158528 - (533*z^2)/27902540178432 - (23*z)/61022855370230784 - 15641/42706235102 30231187456, z, k) - 29192*root(z^6 + (13*z^4)/546912 + (367*z^3)/14353158 528 - (533*z^2)/27902540178432 - (23*z)/61022855370230784 - 15641/42706235 10230231187456, z, k)*x + 441202464*root(z^6 + (13*z^4)/546912 + (367*z^3) /14353158528 - (533*z^2)/27902540178432 - (23*z)/61022855370230784 - 15641 /4270623510230231187456, z, k)^2*x - 775886853888*root(z^6 + (13*z^4)/5469 12 + (367*z^3)/14353158528 - (533*z^2)/27902540178432 - (23*z)/61022855370 230784 - 15641/4270623510230231187456, z, k)^3*x - 138983742784512*root(z^ 6 + (13*z^4)/546912 + (367*z^3)/14353158528 - (533*z^2)/27902540178432 - ( 23*z)/61022855370230784 - 15641/4270623510230231187456, z, k)^4*x - 231366 2762852352*root(z^6 + (13*z^4)/546912 + (367*z^3)/14353158528 - (533*z^2)/ 27902540178432 - (23*z)/61022855370230784 - 15641/4270623510230231187456, z, k)^5*x + 298551744*root(z^6 + (13*z^4)/546912 + (367*z^3)/14353158528 - (533*z^2)/27902540178432 - (23*z)/61022855370230784 - 15641/4270623510230 231187456, z, k)^2 - 563242429440*root(z^6 + (13*z^4)/546912 + (367*z^3)/1 4353158528 - (533*z^2)/27902540178432 - (23*z)/61022855370230784 - 15641/4 270623510230231187456, z, k)^3 + 6435656976384*root(z^6 + (13*z^4)/546912 + (367*z^3)/14353158528 - (533*z^2)/27902540178432 - (23*z)/61022855370230 784 - 15641/4270623510230231187456, z, k)^4 - 56299127229407232*root(z^...
\[ \int \frac {2+3 x}{216+108 x^2+324 x^3+18 x^4+x^6} \, dx=3 \left (\int \frac {x}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \right )+2 \left (\int \frac {1}{x^{6}+18 x^{4}+324 x^{3}+108 x^{2}+216}d x \right ) \] Input:
int((2+3*x)/(x^6+18*x^4+324*x^3+108*x^2+216),x)
Output:
3*int(x/(x**6 + 18*x**4 + 324*x**3 + 108*x**2 + 216),x) + 2*int(1/(x**6 + 18*x**4 + 324*x**3 + 108*x**2 + 216),x)