Integrand size = 71, antiderivative size = 20 \[ \int \frac {1}{4} e^x \left (9+57 x+90 x^2+54 x^3+13 x^4+x^5+\left (18+78 x+72 x^2+22 x^3+2 x^4\right ) \log (3)+\left (9+21 x+9 x^2+x^3\right ) \log ^2(3)\right ) \, dx=\frac {1}{4} e^x x (3+x)^2 (1+x+\log (3))^2 \] Output:
(x+ln(3)+1)^2*(3/2+1/2*x)^2*x*exp(x)
Leaf count is larger than twice the leaf count of optimal. \(54\) vs. \(2(20)=40\).
Time = 5.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.70 \[ \int \frac {1}{4} e^x \left (9+57 x+90 x^2+54 x^3+13 x^4+x^5+\left (18+78 x+72 x^2+22 x^3+2 x^4\right ) \log (3)+\left (9+21 x+9 x^2+x^3\right ) \log ^2(3)\right ) \, dx=\frac {1}{4} e^x x \left (9+x^4+9 \log ^2(3)+x^3 (8+\log (9))+3 x \left (8+2 \log ^2(3)+\log (59049)\right )+x^2 \left (22+\log ^2(3)+\log (4782969)\right )+\log (387420489)\right ) \] Input:
Integrate[(E^x*(9 + 57*x + 90*x^2 + 54*x^3 + 13*x^4 + x^5 + (18 + 78*x + 7 2*x^2 + 22*x^3 + 2*x^4)*Log[3] + (9 + 21*x + 9*x^2 + x^3)*Log[3]^2))/4,x]
Output:
(E^x*x*(9 + x^4 + 9*Log[3]^2 + x^3*(8 + Log[9]) + 3*x*(8 + 2*Log[3]^2 + Lo g[59049]) + x^2*(22 + Log[3]^2 + Log[4782969]) + Log[387420489]))/4
Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(20)=40\).
Time = 0.54 (sec) , antiderivative size = 113, normalized size of antiderivative = 5.65, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {27, 2626, 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.
| \(\displaystyle \int \frac {1}{4} e^x \left (x^5+13 x^4+54 x^3+90 x^2+\left (x^3+9 x^2+21 x+9\right ) \log ^2(3)+\left (2 x^4+22 x^3+72 x^2+78 x+18\right ) \log (3)+57 x+9\right ) \, dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int e^x \left (x^5+13 x^4+54 x^3+90 x^2+57 x+\left (x^3+9 x^2+21 x+9\right ) \log ^2(3)+2 \left (x^4+11 x^3+36 x^2+39 x+9\right ) \log (3)+9\right )dx\) |
| \(\Big \downarrow \) 2626 |
| \(\displaystyle \frac {1}{4} \int \left (e^x x^5+13 e^x x^4+54 e^x x^3+90 e^x x^2+57 e^x x+9 e^x+e^x \left (x^3+9 x^2+21 x+9\right ) \log ^2(3)+2 e^x \left (x^4+11 x^3+36 x^2+39 x+9\right ) \log (3)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (e^x x^5+8 e^x x^4+2 e^x x^4 \log (3)+22 e^x x^3+e^x x^3 \log ^2(3)+14 e^x x^3 \log (3)+24 e^x x^2+6 e^x x^2 \log ^2(3)+30 e^x x^2 \log (3)+9 e^x x+9 e^x x \log ^2(3)+18 e^x x \log (3)\right )\) |
Input:
Int[(E^x*(9 + 57*x + 90*x^2 + 54*x^3 + 13*x^4 + x^5 + (18 + 78*x + 72*x^2 + 22*x^3 + 2*x^4)*Log[3] + (9 + 21*x + 9*x^2 + x^3)*Log[3]^2))/4,x]
Output:
(9*E^x*x + 24*E^x*x^2 + 22*E^x*x^3 + 8*E^x*x^4 + E^x*x^5 + 18*E^x*x*Log[3] + 30*E^x*x^2*Log[3] + 14*E^x*x^3*Log[3] + 2*E^x*x^4*Log[3] + 9*E^x*x*Log[ 3]^2 + 6*E^x*x^2*Log[3]^2 + E^x*x^3*Log[3]^2)/4
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^(v_)*(Px_), x_Symbol] :> Int[ExpandIntegrand[F^v, Px, x], x] /; Fr eeQ[F, x] && PolynomialQ[Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20
| method | result | size |
| gosper | \(\frac {{\mathrm e}^{x} \left (x \ln \left (3\right )+x^{2}+3 \ln \left (3\right )+4 x +3\right )^{2} x}{4}\) | \(24\) |
| norman | \(\left (\frac {\ln \left (3\right )}{2}+2\right ) x^{4} {\mathrm e}^{x}+\left (\frac {\ln \left (3\right )^{2}}{4}+\frac {7 \ln \left (3\right )}{2}+\frac {11}{2}\right ) x^{3} {\mathrm e}^{x}+\left (\frac {3 \ln \left (3\right )^{2}}{2}+\frac {15 \ln \left (3\right )}{2}+6\right ) x^{2} {\mathrm e}^{x}+\left (\frac {9 \ln \left (3\right )^{2}}{4}+\frac {9 \ln \left (3\right )}{2}+\frac {9}{4}\right ) x \,{\mathrm e}^{x}+\frac {x^{5} {\mathrm e}^{x}}{4}\) | \(73\) |
| risch | \(\frac {\left (x^{3} \ln \left (3\right )^{2}+2 x^{4} \ln \left (3\right )+x^{5}+6 x^{2} \ln \left (3\right )^{2}+14 x^{3} \ln \left (3\right )+8 x^{4}+9 x \ln \left (3\right )^{2}+30 x^{2} \ln \left (3\right )+22 x^{3}+18 x \ln \left (3\right )+24 x^{2}+9 x \right ) {\mathrm e}^{x}}{4}\) | \(77\) |
| parallelrisch | \(\frac {{\mathrm e}^{x} \ln \left (3\right )^{2} x^{3}}{4}+\frac {{\mathrm e}^{x} x^{4} \ln \left (3\right )}{2}+\frac {x^{5} {\mathrm e}^{x}}{4}+\frac {3 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x^{2}}{2}+\frac {7 \,{\mathrm e}^{x} \ln \left (3\right ) x^{3}}{2}+2 \,{\mathrm e}^{x} x^{4}+\frac {9 \ln \left (3\right )^{2} {\mathrm e}^{x} x}{4}+\frac {15 \,{\mathrm e}^{x} \ln \left (3\right ) x^{2}}{2}+\frac {11 \,{\mathrm e}^{x} x^{3}}{2}+\frac {9 x \ln \left (3\right ) {\mathrm e}^{x}}{2}+6 \,{\mathrm e}^{x} x^{2}+\frac {9 \,{\mathrm e}^{x} x}{4}\) | \(100\) |
| meijerg | \(\frac {111}{4}+\frac {9 \,{\mathrm e}^{x}}{4}-\left (\frac {\ln \left (3\right )}{2}+\frac {13}{4}\right ) \left (24-\frac {\left (5 x^{4}-20 x^{3}+60 x^{2}-120 x +120\right ) {\mathrm e}^{x}}{5}\right )-\frac {\left (-6 x^{5}+30 x^{4}-120 x^{3}+360 x^{2}-720 x +720\right ) {\mathrm e}^{x}}{24}-\left (-\frac {\ln \left (3\right )^{2}}{4}-\frac {11 \ln \left (3\right )}{2}-\frac {27}{2}\right ) \left (6-\frac {\left (-4 x^{3}+12 x^{2}-24 x +24\right ) {\mathrm e}^{x}}{4}\right )-\left (\frac {9 \ln \left (3\right )^{2}}{4}+18 \ln \left (3\right )+\frac {45}{2}\right ) \left (2-\frac {\left (3 x^{2}-6 x +6\right ) {\mathrm e}^{x}}{3}\right )-\left (-\frac {21 \ln \left (3\right )^{2}}{4}-\frac {39 \ln \left (3\right )}{2}-\frac {57}{4}\right ) \left (1-\frac {\left (2-2 x \right ) {\mathrm e}^{x}}{2}\right )-\frac {9 \ln \left (3\right )^{2} \left (1-{\mathrm e}^{x}\right )}{4}-\frac {9 \ln \left (3\right ) \left (1-{\mathrm e}^{x}\right )}{2}\) | \(182\) |
| default | \(\frac {x^{5} {\mathrm e}^{x}}{4}+2 \,{\mathrm e}^{x} x^{4}+\frac {11 \,{\mathrm e}^{x} x^{3}}{2}+6 \,{\mathrm e}^{x} x^{2}+\frac {9 \,{\mathrm e}^{x} x}{4}+\frac {\ln \left (3\right )^{2} \left ({\mathrm e}^{x} x^{3}-3 \,{\mathrm e}^{x} x^{2}-6 \,{\mathrm e}^{x}+6 \,{\mathrm e}^{x} x \right )}{4}+\frac {9 \,{\mathrm e}^{x} \ln \left (3\right )^{2}}{4}+\frac {9 \ln \left (3\right ) {\mathrm e}^{x}}{2}+\frac {39 \ln \left (3\right ) \left (-{\mathrm e}^{x}+{\mathrm e}^{x} x \right )}{2}+\frac {\ln \left (3\right ) \left ({\mathrm e}^{x} x^{4}-4 \,{\mathrm e}^{x} x^{3}+12 \,{\mathrm e}^{x} x^{2}+24 \,{\mathrm e}^{x}-24 \,{\mathrm e}^{x} x \right )}{2}+18 \ln \left (3\right ) \left ({\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x}-2 \,{\mathrm e}^{x} x \right )+\frac {11 \ln \left (3\right ) \left ({\mathrm e}^{x} x^{3}-3 \,{\mathrm e}^{x} x^{2}-6 \,{\mathrm e}^{x}+6 \,{\mathrm e}^{x} x \right )}{2}+\frac {9 \ln \left (3\right )^{2} \left ({\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x}-2 \,{\mathrm e}^{x} x \right )}{4}+\frac {21 \ln \left (3\right )^{2} \left (-{\mathrm e}^{x}+{\mathrm e}^{x} x \right )}{4}\) | \(209\) |
| parts | \(-\frac {33 \ln \left (3\right ) \left ({\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x}-2 \,{\mathrm e}^{x} x \right )}{2}-36 \ln \left (3\right ) \left (-{\mathrm e}^{x}+{\mathrm e}^{x} x \right )-3 \,{\mathrm e}^{x} \ln \left (3\right )^{2}-15 \ln \left (3\right ) {\mathrm e}^{x}+6 \,{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} x^{4}+\frac {11 \,{\mathrm e}^{x} x^{3}}{2}+\frac {9 \,{\mathrm e}^{x} x}{4}+\frac {x^{5} {\mathrm e}^{x}}{4}+\frac {39 x \ln \left (3\right ) {\mathrm e}^{x}}{2}+\frac {9 \,{\mathrm e}^{x} \ln \left (3\right )^{2} x^{2}}{4}+\frac {{\mathrm e}^{x} \ln \left (3\right )^{2} x^{3}}{4}+\frac {{\mathrm e}^{x} x^{4} \ln \left (3\right )}{2}-\frac {9 \ln \left (3\right )^{2} \left (-{\mathrm e}^{x}+{\mathrm e}^{x} x \right )}{2}-\frac {3 \ln \left (3\right )^{2} \left ({\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x}-2 \,{\mathrm e}^{x} x \right )}{4}-2 \ln \left (3\right ) \left ({\mathrm e}^{x} x^{3}-3 \,{\mathrm e}^{x} x^{2}-6 \,{\mathrm e}^{x}+6 \,{\mathrm e}^{x} x \right )+\frac {21 \ln \left (3\right )^{2} {\mathrm e}^{x} x}{4}+\frac {11 \,{\mathrm e}^{x} \ln \left (3\right ) x^{3}}{2}+18 \,{\mathrm e}^{x} \ln \left (3\right ) x^{2}\) | \(211\) |
Input:
int(1/4*((x^3+9*x^2+21*x+9)*ln(3)^2+(2*x^4+22*x^3+72*x^2+78*x+18)*ln(3)+x^ 5+13*x^4+54*x^3+90*x^2+57*x+9)*exp(x),x,method=_RETURNVERBOSE)
Output:
1/4*exp(x)*(x*ln(3)+x^2+3*ln(3)+4*x+3)^2*x
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (17) = 34\).
Time = 0.10 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.20 \[ \int \frac {1}{4} e^x \left (9+57 x+90 x^2+54 x^3+13 x^4+x^5+\left (18+78 x+72 x^2+22 x^3+2 x^4\right ) \log (3)+\left (9+21 x+9 x^2+x^3\right ) \log ^2(3)\right ) \, dx=\frac {1}{4} \, {\left (x^{5} + 8 \, x^{4} + 22 \, x^{3} + {\left (x^{3} + 6 \, x^{2} + 9 \, x\right )} \log \left (3\right )^{2} + 24 \, x^{2} + 2 \, {\left (x^{4} + 7 \, x^{3} + 15 \, x^{2} + 9 \, x\right )} \log \left (3\right ) + 9 \, x\right )} e^{x} \] Input:
integrate(1/4*((x^3+9*x^2+21*x+9)*log(3)^2+(2*x^4+22*x^3+72*x^2+78*x+18)*l og(3)+x^5+13*x^4+54*x^3+90*x^2+57*x+9)*exp(x),x, algorithm="fricas")
Output:
1/4*(x^5 + 8*x^4 + 22*x^3 + (x^3 + 6*x^2 + 9*x)*log(3)^2 + 24*x^2 + 2*(x^4 + 7*x^3 + 15*x^2 + 9*x)*log(3) + 9*x)*e^x
Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (20) = 40\).
Time = 0.09 (sec) , antiderivative size = 85, normalized size of antiderivative = 4.25 \[ \int \frac {1}{4} e^x \left (9+57 x+90 x^2+54 x^3+13 x^4+x^5+\left (18+78 x+72 x^2+22 x^3+2 x^4\right ) \log (3)+\left (9+21 x+9 x^2+x^3\right ) \log ^2(3)\right ) \, dx=\frac {\left (x^{5} + 2 x^{4} \log {\left (3 \right )} + 8 x^{4} + x^{3} \log {\left (3 \right )}^{2} + 14 x^{3} \log {\left (3 \right )} + 22 x^{3} + 6 x^{2} \log {\left (3 \right )}^{2} + 24 x^{2} + 30 x^{2} \log {\left (3 \right )} + 9 x + 9 x \log {\left (3 \right )}^{2} + 18 x \log {\left (3 \right )}\right ) e^{x}}{4} \] Input:
integrate(1/4*((x**3+9*x**2+21*x+9)*ln(3)**2+(2*x**4+22*x**3+72*x**2+78*x+ 18)*ln(3)+x**5+13*x**4+54*x**3+90*x**2+57*x+9)*exp(x),x)
Output:
(x**5 + 2*x**4*log(3) + 8*x**4 + x**3*log(3)**2 + 14*x**3*log(3) + 22*x**3 + 6*x**2*log(3)**2 + 24*x**2 + 30*x**2*log(3) + 9*x + 9*x*log(3)**2 + 18* x*log(3))*exp(x)/4
Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (17) = 34\).
Time = 0.04 (sec) , antiderivative size = 218, normalized size of antiderivative = 10.90 \[ \int \frac {1}{4} e^x \left (9+57 x+90 x^2+54 x^3+13 x^4+x^5+\left (18+78 x+72 x^2+22 x^3+2 x^4\right ) \log (3)+\left (9+21 x+9 x^2+x^3\right ) \log ^2(3)\right ) \, dx=\frac {1}{4} \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} \log \left (3\right )^{2} + \frac {9}{4} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} \log \left (3\right )^{2} + \frac {21}{4} \, {\left (x - 1\right )} e^{x} \log \left (3\right )^{2} + \frac {1}{2} \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} \log \left (3\right ) + \frac {11}{2} \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} \log \left (3\right ) + 18 \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} \log \left (3\right ) + \frac {39}{2} \, {\left (x - 1\right )} e^{x} \log \left (3\right ) + \frac {9}{4} \, e^{x} \log \left (3\right )^{2} + \frac {1}{4} \, {\left (x^{5} - 5 \, x^{4} + 20 \, x^{3} - 60 \, x^{2} + 120 \, x - 120\right )} e^{x} + \frac {13}{4} \, {\left (x^{4} - 4 \, x^{3} + 12 \, x^{2} - 24 \, x + 24\right )} e^{x} + \frac {27}{2} \, {\left (x^{3} - 3 \, x^{2} + 6 \, x - 6\right )} e^{x} + \frac {45}{2} \, {\left (x^{2} - 2 \, x + 2\right )} e^{x} + \frac {57}{4} \, {\left (x - 1\right )} e^{x} + \frac {9}{2} \, e^{x} \log \left (3\right ) + \frac {9}{4} \, e^{x} \] Input:
integrate(1/4*((x^3+9*x^2+21*x+9)*log(3)^2+(2*x^4+22*x^3+72*x^2+78*x+18)*l og(3)+x^5+13*x^4+54*x^3+90*x^2+57*x+9)*exp(x),x, algorithm="maxima")
Output:
1/4*(x^3 - 3*x^2 + 6*x - 6)*e^x*log(3)^2 + 9/4*(x^2 - 2*x + 2)*e^x*log(3)^ 2 + 21/4*(x - 1)*e^x*log(3)^2 + 1/2*(x^4 - 4*x^3 + 12*x^2 - 24*x + 24)*e^x *log(3) + 11/2*(x^3 - 3*x^2 + 6*x - 6)*e^x*log(3) + 18*(x^2 - 2*x + 2)*e^x *log(3) + 39/2*(x - 1)*e^x*log(3) + 9/4*e^x*log(3)^2 + 1/4*(x^5 - 5*x^4 + 20*x^3 - 60*x^2 + 120*x - 120)*e^x + 13/4*(x^4 - 4*x^3 + 12*x^2 - 24*x + 2 4)*e^x + 27/2*(x^3 - 3*x^2 + 6*x - 6)*e^x + 45/2*(x^2 - 2*x + 2)*e^x + 57/ 4*(x - 1)*e^x + 9/2*e^x*log(3) + 9/4*e^x
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (17) = 34\).
Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.80 \[ \int \frac {1}{4} e^x \left (9+57 x+90 x^2+54 x^3+13 x^4+x^5+\left (18+78 x+72 x^2+22 x^3+2 x^4\right ) \log (3)+\left (9+21 x+9 x^2+x^3\right ) \log ^2(3)\right ) \, dx=\frac {1}{4} \, {\left (x^{5} + 2 \, x^{4} \log \left (3\right ) + x^{3} \log \left (3\right )^{2} + 8 \, x^{4} + 14 \, x^{3} \log \left (3\right ) + 6 \, x^{2} \log \left (3\right )^{2} + 22 \, x^{3} + 30 \, x^{2} \log \left (3\right ) + 9 \, x \log \left (3\right )^{2} + 24 \, x^{2} + 18 \, x \log \left (3\right ) + 9 \, x\right )} e^{x} \] Input:
integrate(1/4*((x^3+9*x^2+21*x+9)*log(3)^2+(2*x^4+22*x^3+72*x^2+78*x+18)*l og(3)+x^5+13*x^4+54*x^3+90*x^2+57*x+9)*exp(x),x, algorithm="giac")
Output:
1/4*(x^5 + 2*x^4*log(3) + x^3*log(3)^2 + 8*x^4 + 14*x^3*log(3) + 6*x^2*log (3)^2 + 22*x^3 + 30*x^2*log(3) + 9*x*log(3)^2 + 24*x^2 + 18*x*log(3) + 9*x )*e^x
Time = 0.06 (sec) , antiderivative size = 62, normalized size of antiderivative = 3.10 \[ \int \frac {1}{4} e^x \left (9+57 x+90 x^2+54 x^3+13 x^4+x^5+\left (18+78 x+72 x^2+22 x^3+2 x^4\right ) \log (3)+\left (9+21 x+9 x^2+x^3\right ) \log ^2(3)\right ) \, dx=\frac {x^5\,{\mathrm {e}}^x}{4}+\frac {x^3\,{\mathrm {e}}^x\,\left (14\,\ln \left (3\right )+{\ln \left (3\right )}^2+22\right )}{4}+\frac {9\,x\,{\mathrm {e}}^x\,{\left (\ln \left (3\right )+1\right )}^2}{4}+\frac {x^4\,{\mathrm {e}}^x\,\left (\ln \left (9\right )+8\right )}{4}+\frac {3\,x^2\,{\mathrm {e}}^x\,\left (\ln \left (3\right )+1\right )\,\left (\ln \left (3\right )+4\right )}{2} \] Input:
int((exp(x)*(57*x + log(3)^2*(21*x + 9*x^2 + x^3 + 9) + log(3)*(78*x + 72* x^2 + 22*x^3 + 2*x^4 + 18) + 90*x^2 + 54*x^3 + 13*x^4 + x^5 + 9))/4,x)
Output:
(x^5*exp(x))/4 + (x^3*exp(x)*(14*log(3) + log(3)^2 + 22))/4 + (9*x*exp(x)* (log(3) + 1)^2)/4 + (x^4*exp(x)*(log(9) + 8))/4 + (3*x^2*exp(x)*(log(3) + 1)*(log(3) + 4))/2
Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 3.40 \[ \int \frac {1}{4} e^x \left (9+57 x+90 x^2+54 x^3+13 x^4+x^5+\left (18+78 x+72 x^2+22 x^3+2 x^4\right ) \log (3)+\left (9+21 x+9 x^2+x^3\right ) \log ^2(3)\right ) \, dx=\frac {e^{x} x \left (\mathrm {log}\left (3\right )^{2} x^{2}+6 \mathrm {log}\left (3\right )^{2} x +9 \mathrm {log}\left (3\right )^{2}+2 \,\mathrm {log}\left (3\right ) x^{3}+14 \,\mathrm {log}\left (3\right ) x^{2}+30 \,\mathrm {log}\left (3\right ) x +18 \,\mathrm {log}\left (3\right )+x^{4}+8 x^{3}+22 x^{2}+24 x +9\right )}{4} \] Input:
int(1/4*((x^3+9*x^2+21*x+9)*log(3)^2+(2*x^4+22*x^3+72*x^2+78*x+18)*log(3)+ x^5+13*x^4+54*x^3+90*x^2+57*x+9)*exp(x),x)
Output:
(e**x*x*(log(3)**2*x**2 + 6*log(3)**2*x + 9*log(3)**2 + 2*log(3)*x**3 + 14 *log(3)*x**2 + 30*log(3)*x + 18*log(3) + x**4 + 8*x**3 + 22*x**2 + 24*x + 9))/4