\(\int \frac {e^{2 x/3} (20+256 x-52 x^2)+e^{2 x/3} (540 x+18 x^2+450 x^3) \log (x)+e^{2 x/3} (972 x^3+162 x^4) \log ^2(x)}{-81+27 \log (4)} \, dx\) [2546]

Optimal result

Integrand size = 75, antiderivative size = 35 \[ \int \frac {e^{2 x/3} \left (20+256 x-52 x^2\right )+e^{2 x/3} \left (540 x+18 x^2+450 x^3\right ) \log (x)+e^{2 x/3} \left (972 x^3+162 x^4\right ) \log ^2(x)}{-81+27 \log (4)} \, dx=2-\frac {e^{2 x/3} \left (5-x+9 x^2 \log (x)\right )^2}{9 (3-\log (4))} \] Output:

2-1/9*(9*x^2*ln(x)-x+5)^2*exp(1/3*x)^2/(3-2*ln(2))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.83 \[ \int \frac {e^{2 x/3} \left (20+256 x-52 x^2\right )+e^{2 x/3} \left (540 x+18 x^2+450 x^3\right ) \log (x)+e^{2 x/3} \left (972 x^3+162 x^4\right ) \log ^2(x)}{-81+27 \log (4)} \, dx=\frac {e^{2 x/3} \left (-5+x-9 x^2 \log (x)\right )^2}{9 (-3+\log (4))} \] Input:

Integrate[(E^((2*x)/3)*(20 + 256*x - 52*x^2) + E^((2*x)/3)*(540*x + 18*x^2 
 + 450*x^3)*Log[x] + E^((2*x)/3)*(972*x^3 + 162*x^4)*Log[x]^2)/(-81 + 27*L 
og[4]),x]
 

Output:

(E^((2*x)/3)*(-5 + x - 9*x^2*Log[x])^2)/(9*(-3 + Log[4]))
 

Rubi [F]

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.

\(\displaystyle \int \frac {e^{2 x/3} \left (-52 x^2+256 x+20\right )+e^{2 x/3} \left (162 x^4+972 x^3\right ) \log ^2(x)+e^{2 x/3} \left (450 x^3+18 x^2+540 x\right ) \log (x)}{27 \log (4)-81} \, dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \left (162 e^{2 x/3} \left (x^4+6 x^3\right ) \log ^2(x)+18 e^{2 x/3} \left (25 x^3+x^2+30 x\right ) \log (x)+4 e^{2 x/3} \left (-13 x^2+64 x+5\right )\right )dx}{27 (3-\log (4))}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {162 \int e^{2 x/3} x^4 \log ^2(x)dx+972 \int e^{2 x/3} x^3 \log ^2(x)dx+\frac {59049 \operatorname {ExpIntegralEi}\left (\frac {2 x}{3}\right )}{4}+675 e^{2 x/3} x^3 \log (x)-\frac {2181}{2} e^{2 x/3} x^2-\frac {6021}{2} e^{2 x/3} x^2 \log (x)+\frac {32685}{4} e^{2 x/3} x-\frac {215913}{8} e^{2 x/3}+\frac {19683}{2} e^{2 x/3} x \log (x)-\frac {59049}{4} e^{2 x/3} \log (x)}{27 (3-\log (4))}\)

Input:

Int[(E^((2*x)/3)*(20 + 256*x - 52*x^2) + E^((2*x)/3)*(540*x + 18*x^2 + 450 
*x^3)*Log[x] + E^((2*x)/3)*(972*x^3 + 162*x^4)*Log[x]^2)/(-81 + 27*Log[4]) 
,x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(66\) vs. \(2(32)=64\).

Time = 0.69 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.91

method	result	size
risch	\(\frac {243 \ln \left (x \right )^{2} {\mathrm e}^{\frac {2 x}{3}} x^{4}}{54 \ln \left (2\right )-81}-\frac {54 x^{2} \left (-5+x \right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )}{54 \ln \left (2\right )-81}+\frac {3 \left (x^{2}-10 x +25\right ) {\mathrm e}^{\frac {2 x}{3}}}{54 \ln \left (2\right )-81}\)	\(67\)
parallelrisch	\(\frac {243 \ln \left (x \right )^{2} {\mathrm e}^{\frac {2 x}{3}} x^{4}-54 \ln \left (x \right ) {\mathrm e}^{\frac {2 x}{3}} x^{3}+270 \ln \left (x \right ) {\mathrm e}^{\frac {2 x}{3}} x^{2}+3 \,{\mathrm e}^{\frac {2 x}{3}} x^{2}-30 \,{\mathrm e}^{\frac {2 x}{3}} x +75 \,{\mathrm e}^{\frac {2 x}{3}}}{54 \ln \left (2\right )-81}\)	\(80\)
orering	\(\frac {9 \left (324 x^{6}+2988 x^{5}+8815 x^{4}-2972 x^{3}-7466 x^{2}+660 x +900\right ) \left (\left (162 x^{4}+972 x^{3}\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )^{2}+\left (450 x^{3}+18 x^{2}+540 x \right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )+\left (-52 x^{2}+256 x +20\right ) {\mathrm e}^{\frac {2 x}{3}}\right )}{2 \left (324 x^{6}+4446 x^{5}+27067 x^{4}+67672 x^{3}-13424 x^{2}-18936 x +1080\right ) \left (54 \ln \left (2\right )-81\right )}-\frac {27 x \left (162 x^{5}+765 x^{4}-223 x^{3}-1114 x^{2}+110 x +300\right ) \left (\left (648 x^{3}+2916 x^{2}\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )^{2}+\frac {2 \left (162 x^{4}+972 x^{3}\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )^{2}}{3}+\frac {2 \left (162 x^{4}+972 x^{3}\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )}{x}+\left (1350 x^{2}+36 x +540\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )+\frac {2 \left (450 x^{3}+18 x^{2}+540 x \right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )}{3}+\frac {\left (450 x^{3}+18 x^{2}+540 x \right ) {\mathrm e}^{\frac {2 x}{3}}}{x}+\left (-104 x +256\right ) {\mathrm e}^{\frac {2 x}{3}}+\frac {2 \left (-52 x^{2}+256 x +20\right ) {\mathrm e}^{\frac {2 x}{3}}}{3}\right )}{2 \left (324 x^{6}+4446 x^{5}+27067 x^{4}+67672 x^{3}-13424 x^{2}-18936 x +1080\right ) \left (54 \ln \left (2\right )-81\right )}+\frac {27 x^{2} \left (81 x^{4}+18 x^{3}-179 x^{2}-20 x +100\right ) \left (\left (1944 x^{2}+5832 x \right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )^{2}+\frac {4 \left (648 x^{3}+2916 x^{2}\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )^{2}}{3}+\frac {4 \left (648 x^{3}+2916 x^{2}\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )}{x}+\frac {4 \left (162 x^{4}+972 x^{3}\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )^{2}}{9}+\frac {8 \left (162 x^{4}+972 x^{3}\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )}{3 x}+\frac {2 \left (162 x^{4}+972 x^{3}\right ) {\mathrm e}^{\frac {2 x}{3}}}{x^{2}}-\frac {2 \left (162 x^{4}+972 x^{3}\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )}{x^{2}}+\left (2700 x +36\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )+\frac {4 \left (1350 x^{2}+36 x +540\right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )}{3}+\frac {2 \left (1350 x^{2}+36 x +540\right ) {\mathrm e}^{\frac {2 x}{3}}}{x}+\frac {4 \left (450 x^{3}+18 x^{2}+540 x \right ) {\mathrm e}^{\frac {2 x}{3}} \ln \left (x \right )}{9}+\frac {4 \left (450 x^{3}+18 x^{2}+540 x \right ) {\mathrm e}^{\frac {2 x}{3}}}{3 x}-\frac {\left (450 x^{3}+18 x^{2}+540 x \right ) {\mathrm e}^{\frac {2 x}{3}}}{x^{2}}-104 \,{\mathrm e}^{\frac {2 x}{3}}+\frac {4 \left (-104 x +256\right ) {\mathrm e}^{\frac {2 x}{3}}}{3}+\frac {4 \left (-52 x^{2}+256 x +20\right ) {\mathrm e}^{\frac {2 x}{3}}}{9}\right )}{2 \left (324 x^{6}+4446 x^{5}+27067 x^{4}+67672 x^{3}-13424 x^{2}-18936 x +1080\right ) \left (54 \ln \left (2\right )-81\right )}\)	\(766\)

Input:

int(((162*x^4+972*x^3)*exp(1/3*x)^2*ln(x)^2+(450*x^3+18*x^2+540*x)*exp(1/3 
*x)^2*ln(x)+(-52*x^2+256*x+20)*exp(1/3*x)^2)/(54*ln(2)-81),x,method=_RETUR 
NVERBOSE)
 

Output:

243/(54*ln(2)-81)*ln(x)^2*exp(2/3*x)*x^4-54/(54*ln(2)-81)*x^2*(-5+x)*exp(2 
/3*x)*ln(x)+3/(54*ln(2)-81)*(x^2-10*x+25)*exp(2/3*x)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.54 \[ \int \frac {e^{2 x/3} \left (20+256 x-52 x^2\right )+e^{2 x/3} \left (540 x+18 x^2+450 x^3\right ) \log (x)+e^{2 x/3} \left (972 x^3+162 x^4\right ) \log ^2(x)}{-81+27 \log (4)} \, dx=\frac {81 \, x^{4} e^{\left (\frac {2}{3} \, x\right )} \log \left (x\right )^{2} - 18 \, {\left (x^{3} - 5 \, x^{2}\right )} e^{\left (\frac {2}{3} \, x\right )} \log \left (x\right ) + {\left (x^{2} - 10 \, x + 25\right )} e^{\left (\frac {2}{3} \, x\right )}}{9 \, {\left (2 \, \log \left (2\right ) - 3\right )}} \] Input:

integrate(((162*x^4+972*x^3)*exp(1/3*x)^2*log(x)^2+(450*x^3+18*x^2+540*x)* 
exp(1/3*x)^2*log(x)+(-52*x^2+256*x+20)*exp(1/3*x)^2)/(54*log(2)-81),x, alg 
orithm="fricas")
 

Output:

1/9*(81*x^4*e^(2/3*x)*log(x)^2 - 18*(x^3 - 5*x^2)*e^(2/3*x)*log(x) + (x^2 
- 10*x + 25)*e^(2/3*x))/(2*log(2) - 3)
 

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.37 \[ \int \frac {e^{2 x/3} \left (20+256 x-52 x^2\right )+e^{2 x/3} \left (540 x+18 x^2+450 x^3\right ) \log (x)+e^{2 x/3} \left (972 x^3+162 x^4\right ) \log ^2(x)}{-81+27 \log (4)} \, dx=\frac {\left (81 x^{4} \log {\left (x \right )}^{2} - 18 x^{3} \log {\left (x \right )} + 90 x^{2} \log {\left (x \right )} + x^{2} - 10 x + 25\right ) e^{\frac {2 x}{3}}}{-27 + 18 \log {\left (2 \right )}} \] Input:

integrate(((162*x**4+972*x**3)*exp(1/3*x)**2*ln(x)**2+(450*x**3+18*x**2+54 
0*x)*exp(1/3*x)**2*ln(x)+(-52*x**2+256*x+20)*exp(1/3*x)**2)/(54*ln(2)-81), 
x)
 

Output:

(81*x**4*log(x)**2 - 18*x**3*log(x) + 90*x**2*log(x) + x**2 - 10*x + 25)*e 
xp(2*x/3)/(-27 + 18*log(2))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (30) = 60\).

Time = 0.10 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.34 \[ \int \frac {e^{2 x/3} \left (20+256 x-52 x^2\right )+e^{2 x/3} \left (540 x+18 x^2+450 x^3\right ) \log (x)+e^{2 x/3} \left (972 x^3+162 x^4\right ) \log ^2(x)}{-81+27 \log (4)} \, dx=\frac {9 \, {\left (9 \, x^{4} \log \left (x\right )^{2} + 3 \, x^{2} - 2 \, {\left (x^{3} - 5 \, x^{2}\right )} \log \left (x\right ) - 24 \, x + 36\right )} e^{\left (\frac {2}{3} \, x\right )} - 13 \, {\left (2 \, x^{2} - 6 \, x + 9\right )} e^{\left (\frac {2}{3} \, x\right )} + 64 \, {\left (2 \, x - 3\right )} e^{\left (\frac {2}{3} \, x\right )} + 10 \, e^{\left (\frac {2}{3} \, x\right )}}{9 \, {\left (2 \, \log \left (2\right ) - 3\right )}} \] Input:

integrate(((162*x^4+972*x^3)*exp(1/3*x)^2*log(x)^2+(450*x^3+18*x^2+540*x)* 
exp(1/3*x)^2*log(x)+(-52*x^2+256*x+20)*exp(1/3*x)^2)/(54*log(2)-81),x, alg 
orithm="maxima")
 

Output:

1/9*(9*(9*x^4*log(x)^2 + 3*x^2 - 2*(x^3 - 5*x^2)*log(x) - 24*x + 36)*e^(2/ 
3*x) - 13*(2*x^2 - 6*x + 9)*e^(2/3*x) + 64*(2*x - 3)*e^(2/3*x) + 10*e^(2/3 
*x))/(2*log(2) - 3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 108 vs. \(2 (30) = 60\).

Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 3.09 \[ \int \frac {e^{2 x/3} \left (20+256 x-52 x^2\right )+e^{2 x/3} \left (540 x+18 x^2+450 x^3\right ) \log (x)+e^{2 x/3} \left (972 x^3+162 x^4\right ) \log ^2(x)}{-81+27 \log (4)} \, dx=\frac {324 \, x^{4} e^{\left (\frac {2}{3} \, x\right )} \log \left (x\right )^{2} + 108 \, x^{2} e^{\left (\frac {2}{3} \, x\right )} + 9 \, {\left (100 \, x^{3} - 446 \, x^{2} + 1458 \, x - 2187\right )} e^{\left (\frac {2}{3} \, x\right )} \log \left (x\right ) - 243 \, {\left (4 \, x^{3} - 18 \, x^{2} + 54 \, x - 81\right )} e^{\left (\frac {2}{3} \, x\right )} \log \left (x\right ) - 4 \, {\left (26 \, x^{2} - 206 \, x + 299\right )} e^{\left (\frac {2}{3} \, x\right )} - 864 \, x e^{\left (\frac {2}{3} \, x\right )} + 1296 \, e^{\left (\frac {2}{3} \, x\right )}}{36 \, {\left (2 \, \log \left (2\right ) - 3\right )}} \] Input:

integrate(((162*x^4+972*x^3)*exp(1/3*x)^2*log(x)^2+(450*x^3+18*x^2+540*x)* 
exp(1/3*x)^2*log(x)+(-52*x^2+256*x+20)*exp(1/3*x)^2)/(54*log(2)-81),x, alg 
orithm="giac")
 

Output:

1/36*(324*x^4*e^(2/3*x)*log(x)^2 + 108*x^2*e^(2/3*x) + 9*(100*x^3 - 446*x^ 
2 + 1458*x - 2187)*e^(2/3*x)*log(x) - 243*(4*x^3 - 18*x^2 + 54*x - 81)*e^( 
2/3*x)*log(x) - 4*(26*x^2 - 206*x + 299)*e^(2/3*x) - 864*x*e^(2/3*x) + 129 
6*e^(2/3*x))/(2*log(2) - 3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{2 x/3} \left (20+256 x-52 x^2\right )+e^{2 x/3} \left (540 x+18 x^2+450 x^3\right ) \log (x)+e^{2 x/3} \left (972 x^3+162 x^4\right ) \log ^2(x)}{-81+27 \log (4)} \, dx=\int \frac {{\mathrm {e}}^{\frac {2\,x}{3}}\,\left (162\,x^4+972\,x^3\right )\,{\ln \left (x\right )}^2+{\mathrm {e}}^{\frac {2\,x}{3}}\,\left (450\,x^3+18\,x^2+540\,x\right )\,\ln \left (x\right )+{\mathrm {e}}^{\frac {2\,x}{3}}\,\left (-52\,x^2+256\,x+20\right )}{54\,\ln \left (2\right )-81} \,d x \] Input:

int((exp((2*x)/3)*(256*x - 52*x^2 + 20) + exp((2*x)/3)*log(x)*(540*x + 18* 
x^2 + 450*x^3) + exp((2*x)/3)*log(x)^2*(972*x^3 + 162*x^4))/(54*log(2) - 8 
1),x)
 

Output:

int((exp((2*x)/3)*(256*x - 52*x^2 + 20) + exp((2*x)/3)*log(x)*(540*x + 18* 
x^2 + 450*x^3) + exp((2*x)/3)*log(x)^2*(972*x^3 + 162*x^4))/(54*log(2) - 8 
1), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.29 \[ \int \frac {e^{2 x/3} \left (20+256 x-52 x^2\right )+e^{2 x/3} \left (540 x+18 x^2+450 x^3\right ) \log (x)+e^{2 x/3} \left (972 x^3+162 x^4\right ) \log ^2(x)}{-81+27 \log (4)} \, dx=\frac {e^{\frac {2 x}{3}} \left (81 \mathrm {log}\left (x \right )^{2} x^{4}-18 \,\mathrm {log}\left (x \right ) x^{3}+90 \,\mathrm {log}\left (x \right ) x^{2}+x^{2}-10 x +25\right )}{18 \,\mathrm {log}\left (2\right )-27} \] Input:

int(((162*x^4+972*x^3)*exp(1/3*x)^2*log(x)^2+(450*x^3+18*x^2+540*x)*exp(1/ 
3*x)^2*log(x)+(-52*x^2+256*x+20)*exp(1/3*x)^2)/(54*log(2)-81),x)
 

Output:

(e**((2*x)/3)*(81*log(x)**2*x**4 - 18*log(x)*x**3 + 90*log(x)*x**2 + x**2 
- 10*x + 25))/(9*(2*log(2) - 3))