\(\int \frac {e^{14 x/5} (-360-165 x-15 x^2)+(45 x+15 x^2+e^{14 x/5} (888 x+447 x^2+42 x^3)) \log (x)+(-360-165 x-15 x^2+(-120 x-15 x^2) \log (x)) \log (8+x)}{(360 x+285 x^2+70 x^3+5 x^4) \log ^2(x)} \, dx\) [89]

Optimal result

Integrand size = 106, antiderivative size = 23 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3 \left (e^{14 x/5}+\log (8+x)\right )}{(3+x) \log (x)} \] Output:

3*(exp(x)^2*exp(2/5*x)^2+ln(x+8))/(3+x)/ln(x)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.49 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3 \left (e^{14 x/5}+\log (8+x)\right )}{(3+x) \log (x)} \] Input:

Integrate[(E^((14*x)/5)*(-360 - 165*x - 15*x^2) + (45*x + 15*x^2 + E^((14* 
x)/5)*(888*x + 447*x^2 + 42*x^3))*Log[x] + (-360 - 165*x - 15*x^2 + (-120* 
x - 15*x^2)*Log[x])*Log[8 + x])/((360*x + 285*x^2 + 70*x^3 + 5*x^4)*Log[x] 
^2),x]
 

Output:

(3*(E^((14*x)/5) + Log[8 + x]))/((3 + 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.

\(\displaystyle \int \frac {e^{14 x/5} \left (-15 x^2-165 x-360\right )+\left (-15 x^2+\left (-15 x^2-120 x\right ) \log (x)-165 x-360\right ) \log (x+8)+\left (15 x^2+e^{14 x/5} \left (42 x^3+447 x^2+888 x\right )+45 x\right ) \log (x)}{\left (5 x^4+70 x^3+285 x^2+360 x\right ) \log ^2(x)} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {e^{14 x/5} \left (-15 x^2-165 x-360\right )+\left (-15 x^2+\left (-15 x^2-120 x\right ) \log (x)-165 x-360\right ) \log (x+8)+\left (15 x^2+e^{14 x/5} \left (42 x^3+447 x^2+888 x\right )+45 x\right ) \log (x)}{x \left (5 x^3+70 x^2+285 x+360\right ) \log ^2(x)}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (-\frac {e^{14 x/5} \left (-15 x^2-165 x-360\right )+\left (-15 x^2+\left (-15 x^2-120 x\right ) \log (x)-165 x-360\right ) \log (x+8)+\left (15 x^2+e^{14 x/5} \left (42 x^3+447 x^2+888 x\right )+45 x\right ) \log (x)}{125 x (x+3) \log ^2(x)}+\frac {e^{14 x/5} \left (-15 x^2-165 x-360\right )+\left (-15 x^2+\left (-15 x^2-120 x\right ) \log (x)-165 x-360\right ) \log (x+8)+\left (15 x^2+e^{14 x/5} \left (42 x^3+447 x^2+888 x\right )+45 x\right ) \log (x)}{125 x (x+8) \log ^2(x)}+\frac {e^{14 x/5} \left (-15 x^2-165 x-360\right )+\left (-15 x^2+\left (-15 x^2-120 x\right ) \log (x)-165 x-360\right ) \log (x+8)+\left (15 x^2+e^{14 x/5} \left (42 x^3+447 x^2+888 x\right )+45 x\right ) \log (x)}{25 x (x+3)^2 \log ^2(x)}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {3 \left (x \log (x) \left (e^{14 x/5} \left (14 x^2+149 x+296\right )+5 (x+3)-5 (x+8) \log (x+8)\right )-5 \left (x^2+11 x+24\right ) \left (e^{14 x/5}+\log (x+8)\right )\right )}{5 x (x+3)^2 (x+8) \log ^2(x)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{5} \int -\frac {5 \left (x^2+11 x+24\right ) \left (\log (x+8)+e^{14 x/5}\right )-x \log (x) \left (5 (x+3)+e^{14 x/5} \left (14 x^2+149 x+296\right )-5 (x+8) \log (x+8)\right )}{x (x+3)^2 (x+8) \log ^2(x)}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {3}{5} \int \frac {5 \left (x^2+11 x+24\right ) \left (\log (x+8)+e^{14 x/5}\right )-x \log (x) \left (5 (x+3)+e^{14 x/5} \left (14 x^2+149 x+296\right )-5 (x+8) \log (x+8)\right )}{x (x+3)^2 (x+8) \log ^2(x)}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {3}{5} \int \left (\frac {5 \left (-\log (x) x^2+\log (x) \log (x+8) x^2+\log (x+8) x^2-3 \log (x) x+8 \log (x) \log (x+8) x+11 \log (x+8) x+24 \log (x+8)\right )}{x (x+3)^2 (x+8) \log ^2(x)}-\frac {e^{14 x/5} \left (14 \log (x) x^2+37 \log (x) x-5 x-15\right )}{x (x+3)^2 \log ^2(x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3}{5} \left (\frac {5}{3} \int \frac {e^{14 x/5}}{x \log ^2(x)}dx-\frac {5}{3} \int \frac {e^{14 x/5}}{(x+3) \log ^2(x)}dx+\frac {5}{3} \int \frac {\log (x+8)}{x \log ^2(x)}dx-\frac {5}{3} \int \frac {\log (x+8)}{(x+3) \log ^2(x)}dx+5 \int \frac {e^{14 x/5}}{(x+3)^2 \log (x)}dx-14 \int \frac {e^{14 x/5}}{(x+3) \log (x)}dx-5 \int \frac {1}{(x+3) (x+8) \log (x)}dx+5 \int \frac {\log (x+8)}{(x+3)^2 \log (x)}dx\right )\)

Input:

Int[(E^((14*x)/5)*(-360 - 165*x - 15*x^2) + (45*x + 15*x^2 + E^((14*x)/5)* 
(888*x + 447*x^2 + 42*x^3))*Log[x] + (-360 - 165*x - 15*x^2 + (-120*x - 15 
*x^2)*Log[x])*Log[8 + x])/((360*x + 285*x^2 + 70*x^3 + 5*x^4)*Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39

\[\frac {3 \ln \left (x +8\right )}{\left (3+x \right ) \ln \left (x \right )}+\frac {3 \,{\mathrm e}^{\frac {14 x}{5}}}{\left (3+x \right ) \ln \left (x \right )}\]

Input:

int((((-15*x^2-120*x)*ln(x)-15*x^2-165*x-360)*ln(x+8)+((42*x^3+447*x^2+888 
*x)*exp(2/5*x)^2*exp(x)^2+15*x^2+45*x)*ln(x)+(-15*x^2-165*x-360)*exp(2/5*x 
)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/ln(x)^2,x)
 

Output:

3/(3+x)/ln(x)*ln(x+8)+3*exp(14/5*x)/(3+x)/ln(x)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{{\left (x + 3\right )} \log \left (x\right )} \] Input:

integrate((((-15*x^2-120*x)*log(x)-15*x^2-165*x-360)*log(x+8)+((42*x^3+447 
*x^2+888*x)*exp(2/5*x)^2*exp(x)^2+15*x^2+45*x)*log(x)+(-15*x^2-165*x-360)* 
exp(2/5*x)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/log(x)^2,x, algorithm= 
"fricas")
 

Output:

3*(e^(14/5*x) + log(x + 8))/((x + 3)*log(x))
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((((-15*x**2-120*x)*ln(x)-15*x**2-165*x-360)*ln(x+8)+((42*x**3+44 
7*x**2+888*x)*exp(2/5*x)**2*exp(x)**2+15*x**2+45*x)*ln(x)+(-15*x**2-165*x- 
360)*exp(2/5*x)**2*exp(x)**2)/(5*x**4+70*x**3+285*x**2+360*x)/ln(x)**2,x)
 

Output:

Exception raised: TypeError >> '>' not supported between instances of 'Pol 
y' and 'int'
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{{\left (x + 3\right )} \log \left (x\right )} \] Input:

integrate((((-15*x^2-120*x)*log(x)-15*x^2-165*x-360)*log(x+8)+((42*x^3+447 
*x^2+888*x)*exp(2/5*x)^2*exp(x)^2+15*x^2+45*x)*log(x)+(-15*x^2-165*x-360)* 
exp(2/5*x)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/log(x)^2,x, algorithm= 
"maxima")
 

Output:

3*(e^(14/5*x) + log(x + 8))/((x + 3)*log(x))
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3 \, {\left (e^{\left (\frac {14}{5} \, x\right )} + \log \left (x + 8\right )\right )}}{x \log \left (x\right ) + 3 \, \log \left (x\right )} \] Input:

integrate((((-15*x^2-120*x)*log(x)-15*x^2-165*x-360)*log(x+8)+((42*x^3+447 
*x^2+888*x)*exp(2/5*x)^2*exp(x)^2+15*x^2+45*x)*log(x)+(-15*x^2-165*x-360)* 
exp(2/5*x)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/log(x)^2,x, algorithm= 
"giac")
 

Output:

3*(e^(14/5*x) + log(x + 8))/(x*log(x) + 3*log(x))
 

Mupad [B] (verification not implemented)

Time = 0.60 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3\,\left ({\mathrm {e}}^{\frac {14\,x}{5}}+\ln \left (x+8\right )\right )}{\ln \left (x\right )\,\left (x+3\right )} \] Input:

int(-(log(x + 8)*(165*x + log(x)*(120*x + 15*x^2) + 15*x^2 + 360) - log(x) 
*(45*x + 15*x^2 + exp(2*x)*exp((4*x)/5)*(888*x + 447*x^2 + 42*x^3)) + exp( 
2*x)*exp((4*x)/5)*(165*x + 15*x^2 + 360))/(log(x)^2*(360*x + 285*x^2 + 70* 
x^3 + 5*x^4)),x)
                                                                                    
                                                                                    
 

Output:

(3*(exp((14*x)/5) + log(x + 8)))/(log(x)*(x + 3))
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {e^{14 x/5} \left (-360-165 x-15 x^2\right )+\left (45 x+15 x^2+e^{14 x/5} \left (888 x+447 x^2+42 x^3\right )\right ) \log (x)+\left (-360-165 x-15 x^2+\left (-120 x-15 x^2\right ) \log (x)\right ) \log (8+x)}{\left (360 x+285 x^2+70 x^3+5 x^4\right ) \log ^2(x)} \, dx=\frac {3 e^{\frac {14 x}{5}}+3 \,\mathrm {log}\left (x +8\right )}{\mathrm {log}\left (x \right ) \left (x +3\right )} \] Input:

int((((-15*x^2-120*x)*log(x)-15*x^2-165*x-360)*log(x+8)+((42*x^3+447*x^2+8 
88*x)*exp(2/5*x)^2*exp(x)^2+15*x^2+45*x)*log(x)+(-15*x^2-165*x-360)*exp(2/ 
5*x)^2*exp(x)^2)/(5*x^4+70*x^3+285*x^2+360*x)/log(x)^2,x)
 

Output:

(3*(e**((14*x)/5) + log(x + 8)))/(log(x)*(x + 3))