\(\int \frac {-2 x^2+3 x^3+3 x^5+e^x (2 x-3 x^3-4 x^4+x^5)+(3 x^2+e^x (-2 x-x^2)) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+(108 x^2-72 x^4+12 x^6) \log (x)+(36 x-12 x^3) \log ^2(x)+4 \log ^3(x)} \, dx\) [1482]

Optimal result

Integrand size = 122, antiderivative size = 32 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\frac {x+\frac {x \left (-e^x+x\right )}{4 \left (-3+x^2-\frac {\log (x)}{x}\right )^2}}{x} \] Output:

(x+1/4*(x-exp(x))*x/(x^2-ln(x)/x-3)^2)/x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.84 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=-\frac {\left (e^x-x\right ) x^2}{4 \left (3 x-x^3+\log (x)\right )^2} \] Input:

Integrate[(-2*x^2 + 3*x^3 + 3*x^5 + E^x*(2*x - 3*x^3 - 4*x^4 + x^5) + (3*x 
^2 + E^x*(-2*x - x^2))*Log[x])/(108*x^3 - 108*x^5 + 36*x^7 - 4*x^9 + (108* 
x^2 - 72*x^4 + 12*x^6)*Log[x] + (36*x - 12*x^3)*Log[x]^2 + 4*Log[x]^3),x]
 

Output:

-1/4*((E^x - x)*x^2)/(3*x - x^3 + 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.

\(\displaystyle \int \frac {3 x^5+3 x^3-2 x^2+\left (3 x^2+e^x \left (-x^2-2 x\right )\right ) \log (x)+e^x \left (x^5-4 x^4-3 x^3+2 x\right )}{-4 x^9+36 x^7-108 x^5+108 x^3+\left (36 x-12 x^3\right ) \log ^2(x)+\left (12 x^6-72 x^4+108 x^2\right ) \log (x)+4 \log ^3(x)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (x \left (3 x^3+3 x-2\right )+e^x \left (x^4-4 x^3-3 x^2+2\right )+3 x \log (x)-e^x (x+2) \log (x)\right )}{4 \left (-x^3+3 x+\log (x)\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int -\frac {x \left (x \left (-3 x^3-3 x+2\right )-e^x \left (x^4-4 x^3-3 x^2+2\right )-3 x \log (x)+e^x (x+2) \log (x)\right )}{\left (-x^3+3 x+\log (x)\right )^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{4} \int \frac {x \left (x \left (-3 x^3-3 x+2\right )-e^x \left (x^4-4 x^3-3 x^2+2\right )-3 x \log (x)+e^x (x+2) \log (x)\right )}{\left (-x^3+3 x+\log (x)\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {1}{4} \int \left (\frac {\left (3 x^3+3 x+3 \log (x)-2\right ) x^2}{\left (x^3-3 x-\log (x)\right )^3}+\frac {e^x \left (x^4-4 x^3-3 x^2-\log (x) x-2 \log (x)+2\right ) x}{\left (x^3-3 x-\log (x)\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} \left (6 \int \frac {x^3}{\left (x^3-3 x-\log (x)\right )^3}dx-6 \int \frac {x^5}{\left (x^3-3 x-\log (x)\right )^3}dx+2 \int \frac {x^2}{\left (x^3-3 x-\log (x)\right )^3}dx+3 \int \frac {x^2}{\left (x^3-3 x-\log (x)\right )^2}dx-\frac {e^x x \left (-x^4+3 x^2+x \log (x)\right )}{\left (-x^3+3 x+\log (x)\right )^3}\right )\)

Input:

Int[(-2*x^2 + 3*x^3 + 3*x^5 + E^x*(2*x - 3*x^3 - 4*x^4 + x^5) + (3*x^2 + E 
^x*(-2*x - x^2))*Log[x])/(108*x^3 - 108*x^5 + 36*x^7 - 4*x^9 + (108*x^2 - 
72*x^4 + 12*x^6)*Log[x] + (36*x - 12*x^3)*Log[x]^2 + 4*Log[x]^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.29 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.78

method	result	size
risch	\(\frac {\left (x -{\mathrm e}^{x}\right ) x^{2}}{4 \left (x^{3}-3 x -\ln \left (x \right )\right )^{2}}\)	\(25\)
parallelrisch	\(\frac {6 x^{3}-6 \,{\mathrm e}^{x} x^{2}}{24 x^{6}-144 x^{4}-48 x^{3} \ln \left (x \right )+216 x^{2}+144 x \ln \left (x \right )+24 \ln \left (x \right )^{2}}\)	\(48\)

Input:

int((((-x^2-2*x)*exp(x)+3*x^2)*ln(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3*x^5+3* 
x^3-2*x^2)/(4*ln(x)^3+(-12*x^3+36*x)*ln(x)^2+(12*x^6-72*x^4+108*x^2)*ln(x) 
-4*x^9+36*x^7-108*x^5+108*x^3),x,method=_RETURNVERBOSE)
 

Output:

1/4*(x-exp(x))*x^2/(x^3-3*x-ln(x))^2
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \] Input:

integrate((((-x^2-2*x)*exp(x)+3*x^2)*log(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3 
*x^5+3*x^3-2*x^2)/(4*log(x)^3+(-12*x^3+36*x)*log(x)^2+(12*x^6-72*x^4+108*x 
^2)*log(x)-4*x^9+36*x^7-108*x^5+108*x^3),x, algorithm="fricas")
 

Output:

1/4*(x^3 - x^2*e^x)/(x^6 - 6*x^4 + 9*x^2 - 2*(x^3 - 3*x)*log(x) + log(x)^2 
)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (24) = 48\).

Time = 0.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 2.50 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\frac {x^{3}}{4 x^{6} - 24 x^{4} + 36 x^{2} + \left (- 8 x^{3} + 24 x\right ) \log {\left (x \right )} + 4 \log {\left (x \right )}^{2}} - \frac {x^{2} e^{x}}{4 x^{6} - 24 x^{4} - 8 x^{3} \log {\left (x \right )} + 36 x^{2} + 24 x \log {\left (x \right )} + 4 \log {\left (x \right )}^{2}} \] Input:

integrate((((-x**2-2*x)*exp(x)+3*x**2)*ln(x)+(x**5-4*x**4-3*x**3+2*x)*exp( 
x)+3*x**5+3*x**3-2*x**2)/(4*ln(x)**3+(-12*x**3+36*x)*ln(x)**2+(12*x**6-72* 
x**4+108*x**2)*ln(x)-4*x**9+36*x**7-108*x**5+108*x**3),x)
 

Output:

x**3/(4*x**6 - 24*x**4 + 36*x**2 + (-8*x**3 + 24*x)*log(x) + 4*log(x)**2) 
- x**2*exp(x)/(4*x**6 - 24*x**4 - 8*x**3*log(x) + 36*x**2 + 24*x*log(x) + 
4*log(x)**2)
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.38 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} + 9 \, x^{2} - 2 \, {\left (x^{3} - 3 \, x\right )} \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \] Input:

integrate((((-x^2-2*x)*exp(x)+3*x^2)*log(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3 
*x^5+3*x^3-2*x^2)/(4*log(x)^3+(-12*x^3+36*x)*log(x)^2+(12*x^6-72*x^4+108*x 
^2)*log(x)-4*x^9+36*x^7-108*x^5+108*x^3),x, algorithm="maxima")
 

Output:

1/4*(x^3 - x^2*e^x)/(x^6 - 6*x^4 + 9*x^2 - 2*(x^3 - 3*x)*log(x) + log(x)^2 
)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.41 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\frac {x^{3} - x^{2} e^{x}}{4 \, {\left (x^{6} - 6 \, x^{4} - 2 \, x^{3} \log \left (x\right ) + 9 \, x^{2} + 6 \, x \log \left (x\right ) + \log \left (x\right )^{2}\right )}} \] Input:

integrate((((-x^2-2*x)*exp(x)+3*x^2)*log(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3 
*x^5+3*x^3-2*x^2)/(4*log(x)^3+(-12*x^3+36*x)*log(x)^2+(12*x^6-72*x^4+108*x 
^2)*log(x)-4*x^9+36*x^7-108*x^5+108*x^3),x, algorithm="giac")
 

Output:

1/4*(x^3 - x^2*e^x)/(x^6 - 6*x^4 - 2*x^3*log(x) + 9*x^2 + 6*x*log(x) + log 
(x)^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\int \frac {{\mathrm {e}}^x\,\left (x^5-4\,x^4-3\,x^3+2\,x\right )-\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (x^2+2\,x\right )-3\,x^2\right )-2\,x^2+3\,x^3+3\,x^5}{{\ln \left (x\right )}^2\,\left (36\,x-12\,x^3\right )+4\,{\ln \left (x\right )}^3+\ln \left (x\right )\,\left (12\,x^6-72\,x^4+108\,x^2\right )+108\,x^3-108\,x^5+36\,x^7-4\,x^9} \,d x \] Input:

int((exp(x)*(2*x - 3*x^3 - 4*x^4 + x^5) - log(x)*(exp(x)*(2*x + x^2) - 3*x 
^2) - 2*x^2 + 3*x^3 + 3*x^5)/(log(x)^2*(36*x - 12*x^3) + 4*log(x)^3 + log( 
x)*(108*x^2 - 72*x^4 + 12*x^6) + 108*x^3 - 108*x^5 + 36*x^7 - 4*x^9),x)
 

Output:

int((exp(x)*(2*x - 3*x^3 - 4*x^4 + x^5) - log(x)*(exp(x)*(2*x + x^2) - 3*x 
^2) - 2*x^2 + 3*x^3 + 3*x^5)/(log(x)^2*(36*x - 12*x^3) + 4*log(x)^3 + log( 
x)*(108*x^2 - 72*x^4 + 12*x^6) + 108*x^3 - 108*x^5 + 36*x^7 - 4*x^9), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.47 \[ \int \frac {-2 x^2+3 x^3+3 x^5+e^x \left (2 x-3 x^3-4 x^4+x^5\right )+\left (3 x^2+e^x \left (-2 x-x^2\right )\right ) \log (x)}{108 x^3-108 x^5+36 x^7-4 x^9+\left (108 x^2-72 x^4+12 x^6\right ) \log (x)+\left (36 x-12 x^3\right ) \log ^2(x)+4 \log ^3(x)} \, dx=\frac {x^{2} \left (-e^{x}+x \right )}{4 \mathrm {log}\left (x \right )^{2}-8 \,\mathrm {log}\left (x \right ) x^{3}+24 \,\mathrm {log}\left (x \right ) x +4 x^{6}-24 x^{4}+36 x^{2}} \] Input:

int((((-x^2-2*x)*exp(x)+3*x^2)*log(x)+(x^5-4*x^4-3*x^3+2*x)*exp(x)+3*x^5+3 
*x^3-2*x^2)/(4*log(x)^3+(-12*x^3+36*x)*log(x)^2+(12*x^6-72*x^4+108*x^2)*lo 
g(x)-4*x^9+36*x^7-108*x^5+108*x^3),x)
 

Output:

(x**2*( - e**x + x))/(4*(log(x)**2 - 2*log(x)*x**3 + 6*log(x)*x + x**6 - 6 
*x**4 + 9*x**2))