\(\int \frac {e (1125+7425 x)+e^x (27 x^2+297 x^3+1089 x^4+1331 x^5)+(225 e+e^x (27 x^2+198 x^3+363 x^4)) \log (x)+e^x (9 x^2+33 x^3) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+(27 x^2+198 x^3+363 x^4) \log (x)+(9 x^2+33 x^3) \log ^2(x)+x^2 \log ^3(x)} \, dx\) [74]

Optimal result

Integrand size = 157, antiderivative size = 25 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=e^x-\frac {9 e}{x \left (2 x+\frac {1}{5} (3+x+\log (x))\right )^2} \] Output:

exp(x)-9/x/(3/5+11/5*x+1/5*ln(x))^2*exp(1)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.76 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=e^x-\frac {225 e}{x (3+11 x+\log (x))^2} \] Input:

Integrate[(E*(1125 + 7425*x) + E^x*(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5 
) + (225*E + E^x*(27*x^2 + 198*x^3 + 363*x^4))*Log[x] + E^x*(9*x^2 + 33*x^ 
3)*Log[x]^2 + E^x*x^2*Log[x]^3)/(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5 + 
(27*x^2 + 198*x^3 + 363*x^4)*Log[x] + (9*x^2 + 33*x^3)*Log[x]^2 + x^2*Log[ 
x]^3),x]
 

Output:

E^x - (225*E)/(x*(3 + 11*x + Log[x])^2)
 

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.

\(\displaystyle \int \frac {e^x x^2 \log ^3(x)+e^x \left (33 x^3+9 x^2\right ) \log ^2(x)+\left (e^x \left (363 x^4+198 x^3+27 x^2\right )+225 e\right ) \log (x)+e^x \left (1331 x^5+1089 x^4+297 x^3+27 x^2\right )+e (7425 x+1125)}{1331 x^5+1089 x^4+297 x^3+27 x^2+x^2 \log ^3(x)+\left (33 x^3+9 x^2\right ) \log ^2(x)+\left (363 x^4+198 x^3+27 x^2\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {e^x x^2 (11 x+3)^3+e^x x^2 \log ^3(x)+3 e^x x^2 (11 x+3) \log ^2(x)+3 \left (e^x x^2 (11 x+3)^2+75 e\right ) \log (x)+225 e (33 x+5)}{x^2 (11 x+\log (x)+3)^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {225 e (33 x+\log (x)+5)}{x^2 (11 x+\log (x)+3)^3}+e^x\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 450 e \int \frac {1}{x^2 (11 x+\log (x)+3)^3}dx+225 e \int \frac {1}{x^2 (11 x+\log (x)+3)^2}dx+4950 e \int \frac {1}{x (11 x+\log (x)+3)^3}dx+e^x\)

Input:

Int[(E*(1125 + 7425*x) + E^x*(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5) + (2 
25*E + E^x*(27*x^2 + 198*x^3 + 363*x^4))*Log[x] + E^x*(9*x^2 + 33*x^3)*Log 
[x]^2 + E^x*x^2*Log[x]^3)/(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5 + (27*x^ 
2 + 198*x^3 + 363*x^4)*Log[x] + (9*x^2 + 33*x^3)*Log[x]^2 + x^2*Log[x]^3), 
x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 4.99 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80

method	result	size
default	\(-\frac {225 \,{\mathrm e}}{x \left (\ln \left (x \right )+11 x +3\right )^{2}}+{\mathrm e}^{x}\)	\(20\)
risch	\(-\frac {225 \,{\mathrm e}}{x \left (\ln \left (x \right )+11 x +3\right )^{2}}+{\mathrm e}^{x}\)	\(20\)
parts	\(-\frac {225 \,{\mathrm e}}{x \left (\ln \left (x \right )+11 x +3\right )^{2}}+{\mathrm e}^{x}\)	\(20\)
parallelrisch	\(-\frac {225 \,{\mathrm e}-66 \,{\mathrm e}^{x} x^{2}-121 \,{\mathrm e}^{x} x^{3}-22 x^{2} {\mathrm e}^{x} \ln \left (x \right )-6 x \,{\mathrm e}^{x} \ln \left (x \right )-9 \,{\mathrm e}^{x} x -x \,{\mathrm e}^{x} \ln \left (x \right )^{2}}{x \left (121 x^{2}+22 x \ln \left (x \right )+\ln \left (x \right )^{2}+66 x +6 \ln \left (x \right )+9\right )}\)	\(80\)

Input:

int((x^2*exp(x)*ln(x)^3+(33*x^3+9*x^2)*exp(x)*ln(x)^2+((363*x^4+198*x^3+27 
*x^2)*exp(x)+225*exp(1))*ln(x)+(1331*x^5+1089*x^4+297*x^3+27*x^2)*exp(x)+( 
7425*x+1125)*exp(1))/(x^2*ln(x)^3+(33*x^3+9*x^2)*ln(x)^2+(363*x^4+198*x^3+ 
27*x^2)*ln(x)+1331*x^5+1089*x^4+297*x^3+27*x^2),x,method=_RETURNVERBOSE)
 

Output:

-225*exp(1)/x/(ln(x)+11*x+3)^2+exp(x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (19) = 38\).

Time = 0.10 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.24 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {x e^{x} \log \left (x\right )^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} e^{x} \log \left (x\right ) + {\left (121 \, x^{3} + 66 \, x^{2} + 9 \, x\right )} e^{x} - 225 \, e}{121 \, x^{3} + x \log \left (x\right )^{2} + 66 \, x^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} \log \left (x\right ) + 9 \, x} \] Input:

integrate((x^2*exp(x)*log(x)^3+(33*x^3+9*x^2)*exp(x)*log(x)^2+((363*x^4+19 
8*x^3+27*x^2)*exp(x)+225*exp(1))*log(x)+(1331*x^5+1089*x^4+297*x^3+27*x^2) 
*exp(x)+(7425*x+1125)*exp(1))/(x^2*log(x)^3+(33*x^3+9*x^2)*log(x)^2+(363*x 
^4+198*x^3+27*x^2)*log(x)+1331*x^5+1089*x^4+297*x^3+27*x^2),x, algorithm=" 
fricas")
 

Output:

(x*e^x*log(x)^2 + 2*(11*x^2 + 3*x)*e^x*log(x) + (121*x^3 + 66*x^2 + 9*x)*e 
^x - 225*e)/(121*x^3 + x*log(x)^2 + 66*x^2 + 2*(11*x^2 + 3*x)*log(x) + 9*x 
)
 

Sympy [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.56 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=e^{x} - \frac {225 e}{121 x^{3} + 66 x^{2} + x \log {\left (x \right )}^{2} + 9 x + \left (22 x^{2} + 6 x\right ) \log {\left (x \right )}} \] Input:

integrate((x**2*exp(x)*ln(x)**3+(33*x**3+9*x**2)*exp(x)*ln(x)**2+((363*x** 
4+198*x**3+27*x**2)*exp(x)+225*exp(1))*ln(x)+(1331*x**5+1089*x**4+297*x**3 
+27*x**2)*exp(x)+(7425*x+1125)*exp(1))/(x**2*ln(x)**3+(33*x**3+9*x**2)*ln( 
x)**2+(363*x**4+198*x**3+27*x**2)*ln(x)+1331*x**5+1089*x**4+297*x**3+27*x* 
*2),x)
 

Output:

exp(x) - 225*E/(121*x**3 + 66*x**2 + x*log(x)**2 + 9*x + (22*x**2 + 6*x)*l 
og(x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (19) = 38\).

Time = 0.09 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.08 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {{\left (121 \, x^{3} + x \log \left (x\right )^{2} + 66 \, x^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} \log \left (x\right ) + 9 \, x\right )} e^{x} - 225 \, e}{121 \, x^{3} + x \log \left (x\right )^{2} + 66 \, x^{2} + 2 \, {\left (11 \, x^{2} + 3 \, x\right )} \log \left (x\right ) + 9 \, x} \] Input:

integrate((x^2*exp(x)*log(x)^3+(33*x^3+9*x^2)*exp(x)*log(x)^2+((363*x^4+19 
8*x^3+27*x^2)*exp(x)+225*exp(1))*log(x)+(1331*x^5+1089*x^4+297*x^3+27*x^2) 
*exp(x)+(7425*x+1125)*exp(1))/(x^2*log(x)^3+(33*x^3+9*x^2)*log(x)^2+(363*x 
^4+198*x^3+27*x^2)*log(x)+1331*x^5+1089*x^4+297*x^3+27*x^2),x, algorithm=" 
maxima")
 

Output:

((121*x^3 + x*log(x)^2 + 66*x^2 + 2*(11*x^2 + 3*x)*log(x) + 9*x)*e^x - 225 
*e)/(121*x^3 + x*log(x)^2 + 66*x^2 + 2*(11*x^2 + 3*x)*log(x) + 9*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (19) = 38\).

Time = 0.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {121 \, x^{3} e^{x} + 22 \, x^{2} e^{x} \log \left (x\right ) + x e^{x} \log \left (x\right )^{2} + 66 \, x^{2} e^{x} + 6 \, x e^{x} \log \left (x\right ) + 9 \, x e^{x} - 225 \, e}{121 \, x^{3} + 22 \, x^{2} \log \left (x\right ) + x \log \left (x\right )^{2} + 66 \, x^{2} + 6 \, x \log \left (x\right ) + 9 \, x} \] Input:

integrate((x^2*exp(x)*log(x)^3+(33*x^3+9*x^2)*exp(x)*log(x)^2+((363*x^4+19 
8*x^3+27*x^2)*exp(x)+225*exp(1))*log(x)+(1331*x^5+1089*x^4+297*x^3+27*x^2) 
*exp(x)+(7425*x+1125)*exp(1))/(x^2*log(x)^3+(33*x^3+9*x^2)*log(x)^2+(363*x 
^4+198*x^3+27*x^2)*log(x)+1331*x^5+1089*x^4+297*x^3+27*x^2),x, algorithm=" 
giac")
 

Output:

(121*x^3*e^x + 22*x^2*e^x*log(x) + x*e^x*log(x)^2 + 66*x^2*e^x + 6*x*e^x*l 
og(x) + 9*x*e^x - 225*e)/(121*x^3 + 22*x^2*log(x) + x*log(x)^2 + 66*x^2 + 
6*x*log(x) + 9*x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\int \frac {{\mathrm {e}}^x\,\left (1331\,x^5+1089\,x^4+297\,x^3+27\,x^2\right )+\ln \left (x\right )\,\left (225\,\mathrm {e}+{\mathrm {e}}^x\,\left (363\,x^4+198\,x^3+27\,x^2\right )\right )+\mathrm {e}\,\left (7425\,x+1125\right )+{\mathrm {e}}^x\,{\ln \left (x\right )}^2\,\left (33\,x^3+9\,x^2\right )+x^2\,{\mathrm {e}}^x\,{\ln \left (x\right )}^3}{\ln \left (x\right )\,\left (363\,x^4+198\,x^3+27\,x^2\right )+{\ln \left (x\right )}^2\,\left (33\,x^3+9\,x^2\right )+x^2\,{\ln \left (x\right )}^3+27\,x^2+297\,x^3+1089\,x^4+1331\,x^5} \,d x \] Input:

int((exp(x)*(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5) + log(x)*(225*exp(1) 
+ exp(x)*(27*x^2 + 198*x^3 + 363*x^4)) + exp(1)*(7425*x + 1125) + exp(x)*l 
og(x)^2*(9*x^2 + 33*x^3) + x^2*exp(x)*log(x)^3)/(log(x)*(27*x^2 + 198*x^3 
+ 363*x^4) + log(x)^2*(9*x^2 + 33*x^3) + x^2*log(x)^3 + 27*x^2 + 297*x^3 + 
 1089*x^4 + 1331*x^5),x)
 

Output:

int((exp(x)*(27*x^2 + 297*x^3 + 1089*x^4 + 1331*x^5) + log(x)*(225*exp(1) 
+ exp(x)*(27*x^2 + 198*x^3 + 363*x^4)) + exp(1)*(7425*x + 1125) + exp(x)*l 
og(x)^2*(9*x^2 + 33*x^3) + x^2*exp(x)*log(x)^3)/(log(x)*(27*x^2 + 198*x^3 
+ 363*x^4) + log(x)^2*(9*x^2 + 33*x^3) + x^2*log(x)^3 + 27*x^2 + 297*x^3 + 
 1089*x^4 + 1331*x^5), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 82, normalized size of antiderivative = 3.28 \[ \int \frac {e (1125+7425 x)+e^x \left (27 x^2+297 x^3+1089 x^4+1331 x^5\right )+\left (225 e+e^x \left (27 x^2+198 x^3+363 x^4\right )\right ) \log (x)+e^x \left (9 x^2+33 x^3\right ) \log ^2(x)+e^x x^2 \log ^3(x)}{27 x^2+297 x^3+1089 x^4+1331 x^5+\left (27 x^2+198 x^3+363 x^4\right ) \log (x)+\left (9 x^2+33 x^3\right ) \log ^2(x)+x^2 \log ^3(x)} \, dx=\frac {e^{x} \mathrm {log}\left (x \right )^{2} x +22 e^{x} \mathrm {log}\left (x \right ) x^{2}+6 e^{x} \mathrm {log}\left (x \right ) x +121 e^{x} x^{3}+66 e^{x} x^{2}+9 e^{x} x -225 e}{x \left (\mathrm {log}\left (x \right )^{2}+22 \,\mathrm {log}\left (x \right ) x +6 \,\mathrm {log}\left (x \right )+121 x^{2}+66 x +9\right )} \] Input:

int((x^2*exp(x)*log(x)^3+(33*x^3+9*x^2)*exp(x)*log(x)^2+((363*x^4+198*x^3+ 
27*x^2)*exp(x)+225*exp(1))*log(x)+(1331*x^5+1089*x^4+297*x^3+27*x^2)*exp(x 
)+(7425*x+1125)*exp(1))/(x^2*log(x)^3+(33*x^3+9*x^2)*log(x)^2+(363*x^4+198 
*x^3+27*x^2)*log(x)+1331*x^5+1089*x^4+297*x^3+27*x^2),x)
 

Output:

(e**x*log(x)**2*x + 22*e**x*log(x)*x**2 + 6*e**x*log(x)*x + 121*e**x*x**3 
+ 66*e**x*x**2 + 9*e**x*x - 225*e)/(x*(log(x)**2 + 22*log(x)*x + 6*log(x) 
+ 121*x**2 + 66*x + 9))