\(\int \frac {e^2 (2 x+2 x^2)+e (52 x+4 x^2) \log (x)+(50 x+2 x^2) \log ^2(x)+(e (96+48 x)+e^2 (2 x+x^2)+e (4 x+2 x^2) \log (x)+(2 x+x^2) \log ^2(x)) \log (2+x)}{e^2 (2 x+x^2)+e (4 x+2 x^2) \log (x)+(2 x+x^2) \log ^2(x)} \, dx\) [1935]

Optimal result

Integrand size = 130, antiderivative size = 22 \[ \int \frac {e^2 \left (2 x+2 x^2\right )+e \left (52 x+4 x^2\right ) \log (x)+\left (50 x+2 x^2\right ) \log ^2(x)+\left (e (96+48 x)+e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log (2+x)}{e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)} \, dx=x-\left (-x-\frac {48 \log (x)}{e+\log (x)}\right ) \log (2+x) \] Output:

x-ln(2+x)*(-x-48/(exp(1)+ln(x))*ln(x))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {e^2 \left (2 x+2 x^2\right )+e \left (52 x+4 x^2\right ) \log (x)+\left (50 x+2 x^2\right ) \log ^2(x)+\left (e (96+48 x)+e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log (2+x)}{e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)} \, dx=x+48 \log (2+x)+\frac {(-48 e+e x+x \log (x)) \log (2+x)}{e+\log (x)} \] Input:

Integrate[(E^2*(2*x + 2*x^2) + E*(52*x + 4*x^2)*Log[x] + (50*x + 2*x^2)*Lo 
g[x]^2 + (E*(96 + 48*x) + E^2*(2*x + x^2) + E*(4*x + 2*x^2)*Log[x] + (2*x 
+ x^2)*Log[x]^2)*Log[2 + x])/(E^2*(2*x + x^2) + E*(4*x + 2*x^2)*Log[x] + ( 
2*x + x^2)*Log[x]^2),x]
 

Output:

x + 48*Log[2 + x] + ((-48*E + E*x + x*Log[x])*Log[2 + x])/(E + 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.

\(\displaystyle \int \frac {e^2 \left (2 x^2+2 x\right )+\left (2 x^2+50 x\right ) \log ^2(x)+\left (e^2 \left (x^2+2 x\right )+\left (x^2+2 x\right ) \log ^2(x)+e \left (2 x^2+4 x\right ) \log (x)+e (48 x+96)\right ) \log (x+2)+e \left (4 x^2+52 x\right ) \log (x)}{e^2 \left (x^2+2 x\right )+\left (x^2+2 x\right ) \log ^2(x)+e \left (2 x^2+4 x\right ) \log (x)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {e^2 \left (2 x^2+2 x\right )+\left (2 x^2+50 x\right ) \log ^2(x)+\left (e^2 \left (x^2+2 x\right )+\left (x^2+2 x\right ) \log ^2(x)+e \left (2 x^2+4 x\right ) \log (x)+e (48 x+96)\right ) \log (x+2)+e \left (4 x^2+52 x\right ) \log (x)}{x (x+2) (\log (x)+e)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 (x+25) \log ^2(x)}{(x+2) (\log (x)+e)^2}+\frac {\left (e^2 x+x \log ^2(x)+2 e x \log (x)+48 e\right ) \log (x+2)}{x (\log (x)+e)^2}+\frac {4 e (x+13) \log (x)}{(x+2) (\log (x)+e)^2}+\frac {2 e^2 (x+1)}{(x+2) (\log (x)+e)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {\log ^2(x) \log (x+2)}{(\log (x)+e)^2}dx+2 e^2 \int \frac {x+1}{(x+2) (\log (x)+e)^2}dx-4 e^2 \int \frac {x+13}{(x+2) (\log (x)+e)^2}dx+2 e^2 \int \frac {x+25}{(x+2) (\log (x)+e)^2}dx+4 e \int \frac {x+13}{(x+2) (\log (x)+e)}dx-4 e \int \frac {x+25}{(x+2) (\log (x)+e)}dx+e^2 \int \frac {\log (x+2)}{(\log (x)+e)^2}dx+48 e \int \frac {\log (x+2)}{x (\log (x)+e)^2}dx+2 e \int \frac {\log (x) \log (x+2)}{(\log (x)+e)^2}dx+2 x+46 \log (x+2)\)

Input:

Int[(E^2*(2*x + 2*x^2) + E*(52*x + 4*x^2)*Log[x] + (50*x + 2*x^2)*Log[x]^2 
 + (E*(96 + 48*x) + E^2*(2*x + x^2) + E*(4*x + 2*x^2)*Log[x] + (2*x + x^2) 
*Log[x]^2)*Log[2 + x])/(E^2*(2*x + x^2) + E*(4*x + 2*x^2)*Log[x] + (2*x + 
x^2)*Log[x]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 6.84 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55

method	result	size
risch	\(\frac {\left (x \,{\mathrm e}+x \ln \left (x \right )-48 \,{\mathrm e}\right ) \ln \left (2+x \right )}{{\mathrm e}+\ln \left (x \right )}+x +48 \ln \left (2+x \right )\)	\(34\)
parallelrisch	\(\frac {\ln \left (2+x \right ) {\mathrm e} x +\ln \left (x \right ) \ln \left (2+x \right ) x +x \,{\mathrm e}+x \ln \left (x \right )+48 \ln \left (2+x \right ) \ln \left (x \right )-4 \,{\mathrm e}-4 \ln \left (x \right )}{{\mathrm e}+\ln \left (x \right )}\)	\(50\)

Input:

int((((x^2+2*x)*ln(x)^2+(2*x^2+4*x)*exp(1)*ln(x)+(x^2+2*x)*exp(1)^2+(48*x+ 
96)*exp(1))*ln(2+x)+(2*x^2+50*x)*ln(x)^2+(4*x^2+52*x)*exp(1)*ln(x)+(2*x^2+ 
2*x)*exp(1)^2)/((x^2+2*x)*ln(x)^2+(2*x^2+4*x)*exp(1)*ln(x)+(x^2+2*x)*exp(1 
)^2),x,method=_RETURNVERBOSE)
 

Output:

(x*exp(1)+x*ln(x)-48*exp(1))/(exp(1)+ln(x))*ln(2+x)+x+48*ln(2+x)
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {e^2 \left (2 x+2 x^2\right )+e \left (52 x+4 x^2\right ) \log (x)+\left (50 x+2 x^2\right ) \log ^2(x)+\left (e (96+48 x)+e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log (2+x)}{e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)} \, dx=\frac {x e + {\left (x e + {\left (x + 48\right )} \log \left (x\right )\right )} \log \left (x + 2\right ) + x \log \left (x\right )}{e + \log \left (x\right )} \] Input:

integrate((((x^2+2*x)*log(x)^2+(2*x^2+4*x)*exp(1)*log(x)+(x^2+2*x)*exp(1)^ 
2+(48*x+96)*exp(1))*log(2+x)+(2*x^2+50*x)*log(x)^2+(4*x^2+52*x)*exp(1)*log 
(x)+(2*x^2+2*x)*exp(1)^2)/((x^2+2*x)*log(x)^2+(2*x^2+4*x)*exp(1)*log(x)+(x 
^2+2*x)*exp(1)^2),x, algorithm="fricas")
 

Output:

(x*e + (x*e + (x + 48)*log(x))*log(x + 2) + x*log(x))/(e + log(x))
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {e^2 \left (2 x+2 x^2\right )+e \left (52 x+4 x^2\right ) \log (x)+\left (50 x+2 x^2\right ) \log ^2(x)+\left (e (96+48 x)+e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log (2+x)}{e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((((x**2+2*x)*ln(x)**2+(2*x**2+4*x)*exp(1)*ln(x)+(x**2+2*x)*exp(1 
)**2+(48*x+96)*exp(1))*ln(2+x)+(2*x**2+50*x)*ln(x)**2+(4*x**2+52*x)*exp(1) 
*ln(x)+(2*x**2+2*x)*exp(1)**2)/((x**2+2*x)*ln(x)**2+(2*x**2+4*x)*exp(1)*ln 
(x)+(x**2+2*x)*exp(1)**2),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {e^2 \left (2 x+2 x^2\right )+e \left (52 x+4 x^2\right ) \log (x)+\left (50 x+2 x^2\right ) \log ^2(x)+\left (e (96+48 x)+e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log (2+x)}{e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)} \, dx=\frac {x e + {\left (x e + {\left (x + 48\right )} \log \left (x\right )\right )} \log \left (x + 2\right ) + x \log \left (x\right )}{e + \log \left (x\right )} \] Input:

integrate((((x^2+2*x)*log(x)^2+(2*x^2+4*x)*exp(1)*log(x)+(x^2+2*x)*exp(1)^ 
2+(48*x+96)*exp(1))*log(2+x)+(2*x^2+50*x)*log(x)^2+(4*x^2+52*x)*exp(1)*log 
(x)+(2*x^2+2*x)*exp(1)^2)/((x^2+2*x)*log(x)^2+(2*x^2+4*x)*exp(1)*log(x)+(x 
^2+2*x)*exp(1)^2),x, algorithm="maxima")
 

Output:

(x*e + (x*e + (x + 48)*log(x))*log(x + 2) + x*log(x))/(e + log(x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (20) = 40\).

Time = 0.13 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {e^2 \left (2 x+2 x^2\right )+e \left (52 x+4 x^2\right ) \log (x)+\left (50 x+2 x^2\right ) \log ^2(x)+\left (e (96+48 x)+e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log (2+x)}{e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)} \, dx=\frac {x e \log \left (x + 2\right ) + x \log \left (x + 2\right ) \log \left (x\right ) + x e + x \log \left (x\right ) + 48 \, \log \left (x + 2\right ) \log \left (x\right )}{e + \log \left (x\right )} \] Input:

integrate((((x^2+2*x)*log(x)^2+(2*x^2+4*x)*exp(1)*log(x)+(x^2+2*x)*exp(1)^ 
2+(48*x+96)*exp(1))*log(2+x)+(2*x^2+50*x)*log(x)^2+(4*x^2+52*x)*exp(1)*log 
(x)+(2*x^2+2*x)*exp(1)^2)/((x^2+2*x)*log(x)^2+(2*x^2+4*x)*exp(1)*log(x)+(x 
^2+2*x)*exp(1)^2),x, algorithm="giac")
 

Output:

(x*e*log(x + 2) + x*log(x + 2)*log(x) + x*e + x*log(x) + 48*log(x + 2)*log 
(x))/(e + log(x))
 

Mupad [B] (verification not implemented)

Time = 3.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \frac {e^2 \left (2 x+2 x^2\right )+e \left (52 x+4 x^2\right ) \log (x)+\left (50 x+2 x^2\right ) \log ^2(x)+\left (e (96+48 x)+e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log (2+x)}{e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)} \, dx=\frac {x\,\mathrm {e}+48\,\ln \left (x+2\right )\,\ln \left (x\right )+x\,\ln \left (x\right )+x\,\ln \left (x+2\right )\,\mathrm {e}+x\,\ln \left (x+2\right )\,\ln \left (x\right )}{\mathrm {e}+\ln \left (x\right )} \] Input:

int((log(x)^2*(50*x + 2*x^2) + exp(2)*(2*x + 2*x^2) + log(x + 2)*(log(x)^2 
*(2*x + x^2) + exp(2)*(2*x + x^2) + exp(1)*(48*x + 96) + exp(1)*log(x)*(4* 
x + 2*x^2)) + exp(1)*log(x)*(52*x + 4*x^2))/(log(x)^2*(2*x + x^2) + exp(2) 
*(2*x + x^2) + exp(1)*log(x)*(4*x + 2*x^2)),x)
 

Output:

(x*exp(1) + 48*log(x + 2)*log(x) + x*log(x) + x*log(x + 2)*exp(1) + x*log( 
x + 2)*log(x))/(exp(1) + log(x))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {e^2 \left (2 x+2 x^2\right )+e \left (52 x+4 x^2\right ) \log (x)+\left (50 x+2 x^2\right ) \log ^2(x)+\left (e (96+48 x)+e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)\right ) \log (2+x)}{e^2 \left (2 x+x^2\right )+e \left (4 x+2 x^2\right ) \log (x)+\left (2 x+x^2\right ) \log ^2(x)} \, dx=\frac {\mathrm {log}\left (x +2\right ) \mathrm {log}\left (x \right ) x +48 \,\mathrm {log}\left (x +2\right ) \mathrm {log}\left (x \right )+\mathrm {log}\left (x +2\right ) e x +\mathrm {log}\left (x \right ) x +e x}{\mathrm {log}\left (x \right )+e} \] Input:

int((((x^2+2*x)*log(x)^2+(2*x^2+4*x)*exp(1)*log(x)+(x^2+2*x)*exp(1)^2+(48* 
x+96)*exp(1))*log(2+x)+(2*x^2+50*x)*log(x)^2+(4*x^2+52*x)*exp(1)*log(x)+(2 
*x^2+2*x)*exp(1)^2)/((x^2+2*x)*log(x)^2+(2*x^2+4*x)*exp(1)*log(x)+(x^2+2*x 
)*exp(1)^2),x)
 

Output:

(log(x + 2)*log(x)*x + 48*log(x + 2)*log(x) + log(x + 2)*e*x + log(x)*x + 
e*x)/(log(x) + e)