\(\int \frac {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} (-15+14 x-6 x^2+x^3)-5 e^{-2+5 x+x^2} x \log (x)}{15 x-5 x^2+e^{-2+5 x+x^2} (9 x^2-6 x^3+x^4)+e^{-2+5 x+x^2} (-15 x+5 x^2) \log (x)} \, dx\) [187]

Optimal result

Integrand size = 112, antiderivative size = 29 \[ \int \frac {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} \left (-15+14 x-6 x^2+x^3\right )-5 e^{-2+5 x+x^2} x \log (x)}{15 x-5 x^2+e^{-2+5 x+x^2} \left (9 x^2-6 x^3+x^4\right )+e^{-2+5 x+x^2} \left (-15 x+5 x^2\right ) \log (x)} \, dx=\log \left (\frac {1}{4} \left (x+\frac {5 \left (-e^{2-x (5+x)}+\log (x)\right )}{-3+x}\right )\right ) \] Output:

ln(1/4*x+5/4*(ln(x)-1/exp((5+x)*x-2))/(-3+x))
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} \left (-15+14 x-6 x^2+x^3\right )-5 e^{-2+5 x+x^2} x \log (x)}{15 x-5 x^2+e^{-2+5 x+x^2} \left (9 x^2-6 x^3+x^4\right )+e^{-2+5 x+x^2} \left (-15 x+5 x^2\right ) \log (x)} \, dx=-\frac {1}{4} (5+2 x)^2-\log (3-x)+\log \left (5 e^2-e^{x (5+x)} (-3+x) x-5 e^{x (5+x)} \log (x)\right ) \] Input:

Integrate[(-70*x - 5*x^2 + 10*x^3 + E^(-2 + 5*x + x^2)*(-15 + 14*x - 6*x^2 
 + x^3) - 5*E^(-2 + 5*x + x^2)*x*Log[x])/(15*x - 5*x^2 + E^(-2 + 5*x + x^2 
)*(9*x^2 - 6*x^3 + x^4) + E^(-2 + 5*x + x^2)*(-15*x + 5*x^2)*Log[x]),x]
 

Output:

-1/4*(5 + 2*x)^2 - Log[3 - x] + Log[5*E^2 - E^(x*(5 + x))*(-3 + x)*x - 5*E 
^(x*(5 + x))*Log[x]]
 

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.

\(\displaystyle \int \frac {10 x^3-5 x^2-5 e^{x^2+5 x-2} x \log (x)+e^{x^2+5 x-2} \left (x^3-6 x^2+14 x-15\right )-70 x}{-5 x^2+e^{x^2+5 x-2} \left (5 x^2-15 x\right ) \log (x)+e^{x^2+5 x-2} \left (x^4-6 x^3+9 x^2\right )+15 x} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^2 \left (10 x^3-5 x^2-5 e^{x^2+5 x-2} x \log (x)+e^{x^2+5 x-2} \left (x^3-6 x^2+14 x-15\right )-70 x\right )}{(3-x) x \left (-e^{x (x+5)} x^2+3 e^{x (x+5)} x-5 e^{x (x+5)} \log (x)+5 e^2\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle e^2 \int -\frac {-10 x^3+5 x^2+5 e^{x^2+5 x-2} \log (x) x+70 x+e^{x^2+5 x-2} \left (-x^3+6 x^2-14 x+15\right )}{(3-x) x \left (-e^{x (x+5)} x^2+3 e^{x (x+5)} x-5 e^{x (x+5)} \log (x)+5 e^2\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -e^2 \int \frac {-10 x^3+5 x^2+5 e^{x^2+5 x-2} \log (x) x+70 x+e^{x^2+5 x-2} \left (-x^3+6 x^2-14 x+15\right )}{(3-x) x \left (-e^{x (x+5)} x^2+3 e^{x (x+5)} x-5 e^{x (x+5)} \log (x)+5 e^2\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -e^2 \int \left (\frac {e^{x (x+5)} x^3+10 e^2 x^3-6 e^{x (x+5)} x^2-5 e^2 x^2+14 e^{x (x+5)} x-5 e^{x (x+5)} \log (x) x-70 e^2 x-15 e^{x (x+5)}}{3 e^2 (x-3) \left (-e^{x (x+5)} x^2+3 e^{x (x+5)} x-5 e^{x (x+5)} \log (x)+5 e^2\right )}-\frac {e^{x (x+5)} x^3+10 e^2 x^3-6 e^{x (x+5)} x^2-5 e^2 x^2+14 e^{x (x+5)} x-5 e^{x (x+5)} \log (x) x-70 e^2 x-15 e^{x (x+5)}}{3 e^2 x \left (-e^{x (x+5)} x^2+3 e^{x (x+5)} x-5 e^{x (x+5)} \log (x)+5 e^2\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -e^2 \int \frac {-5 e^2 x \left (2 x^2-x-14\right )-e^{x (x+5)} \left (x^3-6 x^2+14 x-15\right )+5 e^{x (x+5)} x \log (x)}{e^2 (3-x) x \left (-e^{x (x+5)} (x-3) x-5 e^{x (x+5)} \log (x)+5 e^2\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\int \frac {5 e^2 x \left (-2 x^2+x+14\right )+e^{x (x+5)} \left (-x^3+6 x^2-14 x+15\right )+5 e^{x (x+5)} x \log (x)}{(3-x) x \left (e^{x (x+5)} (3-x) x-5 e^{x (x+5)} \log (x)+5 e^2\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\int \left (\frac {-x^3+6 x^2+5 \log (x) x-14 x+15}{(x-3) x \left (x^2-3 x+5 \log (x)\right )}+\frac {5 e^2 \left (2 x^4-x^3+10 \log (x) x^2-13 x^2+25 \log (x) x-3 x+5\right )}{x \left (x^2-3 x+5 \log (x)\right ) \left (-e^{x (x+5)} x^2+3 e^{x (x+5)} x-5 e^{x (x+5)} \log (x)+5 e^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -15 e^2 \int \frac {1}{\left (x^2-3 x+5 \log (x)\right ) \left (e^{x (x+5)} x^2-3 e^{x (x+5)} x+5 e^{x (x+5)} \log (x)-5 e^2\right )}dx+25 e^2 \int \frac {1}{x \left (x^2-3 x+5 \log (x)\right ) \left (e^{x (x+5)} x^2-3 e^{x (x+5)} x+5 e^{x (x+5)} \log (x)-5 e^2\right )}dx-65 e^2 \int \frac {x}{\left (x^2-3 x+5 \log (x)\right ) \left (e^{x (x+5)} x^2-3 e^{x (x+5)} x+5 e^{x (x+5)} \log (x)-5 e^2\right )}dx-5 e^2 \int \frac {x^2}{\left (x^2-3 x+5 \log (x)\right ) \left (e^{x (x+5)} x^2-3 e^{x (x+5)} x+5 e^{x (x+5)} \log (x)-5 e^2\right )}dx+125 e^2 \int \frac {\log (x)}{\left (x^2-3 x+5 \log (x)\right ) \left (e^{x (x+5)} x^2-3 e^{x (x+5)} x+5 e^{x (x+5)} \log (x)-5 e^2\right )}dx+50 e^2 \int \frac {x \log (x)}{\left (x^2-3 x+5 \log (x)\right ) \left (e^{x (x+5)} x^2-3 e^{x (x+5)} x+5 e^{x (x+5)} \log (x)-5 e^2\right )}dx+10 e^2 \int \frac {x^3}{\left (x^2-3 x+5 \log (x)\right ) \left (e^{x (x+5)} x^2-3 e^{x (x+5)} x+5 e^{x (x+5)} \log (x)-5 e^2\right )}dx+\log \left (-x^2+3 x-5 \log (x)\right )-\log (3-x)\)

Input:

Int[(-70*x - 5*x^2 + 10*x^3 + E^(-2 + 5*x + x^2)*(-15 + 14*x - 6*x^2 + x^3 
) - 5*E^(-2 + 5*x + x^2)*x*Log[x])/(15*x - 5*x^2 + E^(-2 + 5*x + x^2)*(9*x 
^2 - 6*x^3 + x^4) + E^(-2 + 5*x + x^2)*(-15*x + 5*x^2)*Log[x]),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14

method	result	size
risch	\(-\ln \left (-3+x \right )+\ln \left (\ln \left (x \right )+\frac {x^{2}}{5}-\frac {3 x}{5}-{\mathrm e}^{-x^{2}-5 x +2}\right )\)	\(33\)
norman	\(-5 x -x^{2}-\ln \left (-3+x \right )+\ln \left (x^{2} {\mathrm e}^{x^{2}+5 x -2}-3 \,{\mathrm e}^{x^{2}+5 x -2} x +5 \,{\mathrm e}^{x^{2}+5 x -2} \ln \left (x \right )-5\right )\)	\(57\)
parallelrisch	\(-5 x -x^{2}-\ln \left (-3+x \right )+\ln \left (x^{2} {\mathrm e}^{x^{2}+5 x -2}-3 \,{\mathrm e}^{x^{2}+5 x -2} x +5 \,{\mathrm e}^{x^{2}+5 x -2} \ln \left (x \right )-5\right )\)	\(57\)

Input:

int((-5*x*exp(x^2+5*x-2)*ln(x)+(x^3-6*x^2+14*x-15)*exp(x^2+5*x-2)+10*x^3-5 
*x^2-70*x)/((5*x^2-15*x)*exp(x^2+5*x-2)*ln(x)+(x^4-6*x^3+9*x^2)*exp(x^2+5* 
x-2)-5*x^2+15*x),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.79 \[ \int \frac {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} \left (-15+14 x-6 x^2+x^3\right )-5 e^{-2+5 x+x^2} x \log (x)}{15 x-5 x^2+e^{-2+5 x+x^2} \left (9 x^2-6 x^3+x^4\right )+e^{-2+5 x+x^2} \left (-15 x+5 x^2\right ) \log (x)} \, dx=\log \left ({\left ({\left (x^{2} - 3 \, x\right )} e^{\left (x^{2} + 5 \, x - 2\right )} + 5 \, e^{\left (x^{2} + 5 \, x - 2\right )} \log \left (x\right ) - 5\right )} e^{\left (-x^{2} - 5 \, x + 2\right )}\right ) - \log \left (x - 3\right ) \] Input:

integrate((-5*x*exp(x^2+5*x-2)*log(x)+(x^3-6*x^2+14*x-15)*exp(x^2+5*x-2)+1 
0*x^3-5*x^2-70*x)/((5*x^2-15*x)*exp(x^2+5*x-2)*log(x)+(x^4-6*x^3+9*x^2)*ex 
p(x^2+5*x-2)-5*x^2+15*x),x, algorithm="fricas")
 

Output:

log(((x^2 - 3*x)*e^(x^2 + 5*x - 2) + 5*e^(x^2 + 5*x - 2)*log(x) - 5)*e^(-x 
^2 - 5*x + 2)) - log(x - 3)
 

Sympy [A] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.76 \[ \int \frac {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} \left (-15+14 x-6 x^2+x^3\right )-5 e^{-2+5 x+x^2} x \log (x)}{15 x-5 x^2+e^{-2+5 x+x^2} \left (9 x^2-6 x^3+x^4\right )+e^{-2+5 x+x^2} \left (-15 x+5 x^2\right ) \log (x)} \, dx=- x^{2} - 5 x - \log {\left (x - 3 \right )} + \log {\left (e^{x^{2} + 5 x - 2} - \frac {5}{x^{2} - 3 x + 5 \log {\left (x \right )}} \right )} + \log {\left (\frac {x^{2}}{5} - \frac {3 x}{5} + \log {\left (x \right )} \right )} \] Input:

integrate((-5*x*exp(x**2+5*x-2)*ln(x)+(x**3-6*x**2+14*x-15)*exp(x**2+5*x-2 
)+10*x**3-5*x**2-70*x)/((5*x**2-15*x)*exp(x**2+5*x-2)*ln(x)+(x**4-6*x**3+9 
*x**2)*exp(x**2+5*x-2)-5*x**2+15*x),x)
 

Output:

-x**2 - 5*x - log(x - 3) + log(exp(x**2 + 5*x - 2) - 5/(x**2 - 3*x + 5*log 
(x))) + log(x**2/5 - 3*x/5 + log(x))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 68 vs. \(2 (26) = 52\).

Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 2.34 \[ \int \frac {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} \left (-15+14 x-6 x^2+x^3\right )-5 e^{-2+5 x+x^2} x \log (x)}{15 x-5 x^2+e^{-2+5 x+x^2} \left (9 x^2-6 x^3+x^4\right )+e^{-2+5 x+x^2} \left (-15 x+5 x^2\right ) \log (x)} \, dx=-x^{2} + \log \left (\frac {1}{5} \, x^{2} - \frac {3}{5} \, x + \log \left (x\right )\right ) - \log \left (x - 3\right ) + \log \left (\frac {{\left ({\left (x^{2} - 3 \, x + 5 \, \log \left (x\right )\right )} e^{\left (x^{2} + 5 \, x\right )} - 5 \, e^{2}\right )} e^{\left (-5 \, x\right )}}{x^{2} - 3 \, x + 5 \, \log \left (x\right )}\right ) \] Input:

integrate((-5*x*exp(x^2+5*x-2)*log(x)+(x^3-6*x^2+14*x-15)*exp(x^2+5*x-2)+1 
0*x^3-5*x^2-70*x)/((5*x^2-15*x)*exp(x^2+5*x-2)*log(x)+(x^4-6*x^3+9*x^2)*ex 
p(x^2+5*x-2)-5*x^2+15*x),x, algorithm="maxima")
 

Output:

-x^2 + log(1/5*x^2 - 3/5*x + log(x)) - log(x - 3) + log(((x^2 - 3*x + 5*lo 
g(x))*e^(x^2 + 5*x) - 5*e^2)*e^(-5*x)/(x^2 - 3*x + 5*log(x)))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (26) = 52\).

Time = 0.14 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.93 \[ \int \frac {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} \left (-15+14 x-6 x^2+x^3\right )-5 e^{-2+5 x+x^2} x \log (x)}{15 x-5 x^2+e^{-2+5 x+x^2} \left (9 x^2-6 x^3+x^4\right )+e^{-2+5 x+x^2} \left (-15 x+5 x^2\right ) \log (x)} \, dx=-x^{2} - 5 \, x + \log \left (-x^{2} e^{\left (x^{2} + 5 \, x\right )} + 3 \, x e^{\left (x^{2} + 5 \, x\right )} - 5 \, e^{\left (x^{2} + 5 \, x\right )} \log \left (x\right ) + 5 \, e^{2}\right ) + \log \left (x^{2} - 3 \, x + 5 \, \log \left (x\right )\right ) - \log \left (-x^{2} + 3 \, x - 5 \, \log \left (x\right )\right ) - \log \left (x - 3\right ) \] Input:

integrate((-5*x*exp(x^2+5*x-2)*log(x)+(x^3-6*x^2+14*x-15)*exp(x^2+5*x-2)+1 
0*x^3-5*x^2-70*x)/((5*x^2-15*x)*exp(x^2+5*x-2)*log(x)+(x^4-6*x^3+9*x^2)*ex 
p(x^2+5*x-2)-5*x^2+15*x),x, algorithm="giac")
 

Output:

-x^2 - 5*x + log(-x^2*e^(x^2 + 5*x) + 3*x*e^(x^2 + 5*x) - 5*e^(x^2 + 5*x)* 
log(x) + 5*e^2) + log(x^2 - 3*x + 5*log(x)) - log(-x^2 + 3*x - 5*log(x)) - 
 log(x - 3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} \left (-15+14 x-6 x^2+x^3\right )-5 e^{-2+5 x+x^2} x \log (x)}{15 x-5 x^2+e^{-2+5 x+x^2} \left (9 x^2-6 x^3+x^4\right )+e^{-2+5 x+x^2} \left (-15 x+5 x^2\right ) \log (x)} \, dx=\int -\frac {70\,x-{\mathrm {e}}^{x^2+5\,x-2}\,\left (x^3-6\,x^2+14\,x-15\right )+5\,x^2-10\,x^3+5\,x\,{\mathrm {e}}^{x^2+5\,x-2}\,\ln \left (x\right )}{15\,x+{\mathrm {e}}^{x^2+5\,x-2}\,\left (x^4-6\,x^3+9\,x^2\right )-5\,x^2-{\mathrm {e}}^{x^2+5\,x-2}\,\ln \left (x\right )\,\left (15\,x-5\,x^2\right )} \,d x \] Input:

int(-(70*x - exp(5*x + x^2 - 2)*(14*x - 6*x^2 + x^3 - 15) + 5*x^2 - 10*x^3 
 + 5*x*exp(5*x + x^2 - 2)*log(x))/(15*x + exp(5*x + x^2 - 2)*(9*x^2 - 6*x^ 
3 + x^4) - 5*x^2 - exp(5*x + x^2 - 2)*log(x)*(15*x - 5*x^2)),x)
 

Output:

int(-(70*x - exp(5*x + x^2 - 2)*(14*x - 6*x^2 + x^3 - 15) + 5*x^2 - 10*x^3 
 + 5*x*exp(5*x + x^2 - 2)*log(x))/(15*x + exp(5*x + x^2 - 2)*(9*x^2 - 6*x^ 
3 + x^4) - 5*x^2 - exp(5*x + x^2 - 2)*log(x)*(15*x - 5*x^2)), x)
 

Reduce [B] (verification not implemented)

Time = 67.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.07 \[ \int \frac {-70 x-5 x^2+10 x^3+e^{-2+5 x+x^2} \left (-15+14 x-6 x^2+x^3\right )-5 e^{-2+5 x+x^2} x \log (x)}{15 x-5 x^2+e^{-2+5 x+x^2} \left (9 x^2-6 x^3+x^4\right )+e^{-2+5 x+x^2} \left (-15 x+5 x^2\right ) \log (x)} \, dx=\mathrm {log}\left (5 e^{x^{2}+5 x} \mathrm {log}\left (x \right )+e^{x^{2}+5 x} x^{2}-3 e^{x^{2}+5 x} x -5 e^{2}\right )-\mathrm {log}\left (x -3\right )-x^{2}-5 x \] Input:

int((-5*x*exp(x^2+5*x-2)*log(x)+(x^3-6*x^2+14*x-15)*exp(x^2+5*x-2)+10*x^3- 
5*x^2-70*x)/((5*x^2-15*x)*exp(x^2+5*x-2)*log(x)+(x^4-6*x^3+9*x^2)*exp(x^2+ 
5*x-2)-5*x^2+15*x),x)
 

Output:

log(5*e**(x**2 + 5*x)*log(x) + e**(x**2 + 5*x)*x**2 - 3*e**(x**2 + 5*x)*x 
- 5*e**2) - log(x - 3) - x**2 - 5*x