\(\int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 (16 x+10 x^2+6 x^3)+e^2 (14+46 x+30 x^2+16 x^3+6 x^4)+(-26 x+2 e^4 x-2 x^2+e^2 (2+6 x+2 x^2)) \log (25 e^x x)-2 x \log ^2(25 e^x x)}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx\) [531]

Optimal result

Integrand size = 151, antiderivative size = 24 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=\left (x+\frac {7+2 x+\log \left (25 e^x x\right )}{e^2+x}\right )^2 \] Output:

(x+(7+ln(25*exp(x)*x)+2*x)/(x+exp(2)))^2
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(113\) vs. \(2(24)=48\).

Time = 0.05 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.71 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=\frac {49-23 e^4+6 e^6+28 x-32 e^2 x+14 e^4 x-5 x^2+14 e^2 x^2+e^4 x^2+6 x^3+2 e^2 x^3+x^4+2 \left (e^2+x\right )^2 \log (x)-2 \left (-7+e^4-2 x+e^2 x\right ) \log \left (25 e^x x\right )+\log ^2\left (25 e^x x\right )}{\left (e^2+x\right )^2} \] Input:

Integrate[(-84*x - 10*x^2 + 2*E^6*x^2 + 6*x^3 + 6*x^4 + 2*x^5 + E^4*(16*x 
+ 10*x^2 + 6*x^3) + E^2*(14 + 46*x + 30*x^2 + 16*x^3 + 6*x^4) + (-26*x + 2 
*E^4*x - 2*x^2 + E^2*(2 + 6*x + 2*x^2))*Log[25*E^x*x] - 2*x*Log[25*E^x*x]^ 
2)/(E^6*x + 3*E^4*x^2 + 3*E^2*x^3 + x^4),x]
 

Output:

(49 - 23*E^4 + 6*E^6 + 28*x - 32*E^2*x + 14*E^4*x - 5*x^2 + 14*E^2*x^2 + E 
^4*x^2 + 6*x^3 + 2*E^2*x^3 + x^4 + 2*(E^2 + x)^2*Log[x] - 2*(-7 + E^4 - 2* 
x + E^2*x)*Log[25*E^x*x] + Log[25*E^x*x]^2)/(E^2 + 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.

\(\displaystyle \int \frac {2 x^5+6 x^4+6 x^3+2 e^6 x^2-10 x^2+\left (-2 x^2+e^2 \left (2 x^2+6 x+2\right )+2 e^4 x-26 x\right ) \log \left (25 e^x x\right )+e^4 \left (6 x^3+10 x^2+16 x\right )+e^2 \left (6 x^4+16 x^3+30 x^2+46 x+14\right )-84 x-2 x \log ^2\left (25 e^x x\right )}{x^4+3 e^2 x^3+3 e^4 x^2+e^6 x} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {2 x^5+6 x^4+6 x^3+\left (2 e^6-10\right ) x^2+\left (-2 x^2+e^2 \left (2 x^2+6 x+2\right )+2 e^4 x-26 x\right ) \log \left (25 e^x x\right )+e^4 \left (6 x^3+10 x^2+16 x\right )+e^2 \left (6 x^4+16 x^3+30 x^2+46 x+14\right )-84 x-2 x \log ^2\left (25 e^x x\right )}{x^4+3 e^2 x^3+3 e^4 x^2+e^6 x}dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {2 x^5+6 x^4+6 x^3+\left (2 e^6-10\right ) x^2+\left (-2 x^2+e^2 \left (2 x^2+6 x+2\right )+2 e^4 x-26 x\right ) \log \left (25 e^x x\right )+e^4 \left (6 x^3+10 x^2+16 x\right )+e^2 \left (6 x^4+16 x^3+30 x^2+46 x+14\right )-84 x-2 x \log ^2\left (25 e^x x\right )}{x \left (x^3+3 e^2 x^2+3 e^4 x+e^6\right )}dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {2 x^5+6 x^4+6 x^3+\left (2 e^6-10\right ) x^2+\left (-2 x^2+e^2 \left (2 x^2+6 x+2\right )+2 e^4 x-26 x\right ) \log \left (25 e^x x\right )+e^4 \left (6 x^3+10 x^2+16 x\right )+e^2 \left (6 x^4+16 x^3+30 x^2+46 x+14\right )-84 x-2 x \log ^2\left (25 e^x x\right )}{x \left (x+e^2\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4}{\left (x+e^2\right )^3}+\frac {6 x^3}{\left (x+e^2\right )^3}+\frac {6 x^2}{\left (x+e^2\right )^3}+\frac {2 e^4 \left (3 x^2+5 x+8\right )}{\left (x+e^2\right )^3}+\frac {2 \left (-\left (\left (1-e^2\right ) x^2\right )-\left (13-3 e^2-e^4\right ) x+e^2\right ) \log \left (25 e^x x\right )}{\left (x+e^2\right )^3 x}+\frac {2 e^2 \left (3 x^4+8 x^3+15 x^2+23 x+7\right )}{\left (x+e^2\right )^3 x}+\frac {2 \left (e^6-5\right ) x}{\left (x+e^2\right )^3}-\frac {84}{\left (x+e^2\right )^3}-\frac {2 \log ^2\left (25 e^x x\right )}{\left (x+e^2\right )^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 \int \frac {\log ^2\left (25 e^x x\right )}{\left (x+e^2\right )^3}dx+\frac {2 \int \frac {\log \left (25 e^x x\right )}{x}dx}{e^4}-\frac {2 \int \frac {\log \left (25 e^x x\right )}{x+e^2}dx}{e^4}-\frac {\left (5-e^6\right ) x^2}{e^2 \left (x+e^2\right )^2}+x^2+6 x+\frac {2 \left (7-15 e^4+16 e^6-9 e^8\right )}{e^2 \left (x+e^2\right )}+\frac {2 \left (1-\frac {1}{e^2}\right ) \left (7-2 e^2\right )}{x+e^2}-\frac {2 e^4 \left (5-6 e^2\right )}{x+e^2}+\frac {8 e^6}{x+e^2}-\frac {18 e^4}{x+e^2}+\frac {12 e^2}{x+e^2}+\frac {7-23 e^2+15 e^4-8 e^6+3 e^8}{\left (x+e^2\right )^2}-\frac {e^4 \left (8-5 e^2+3 e^4\right )}{\left (x+e^2\right )^2}-\frac {e^8}{\left (x+e^2\right )^2}+\frac {3 e^6}{\left (x+e^2\right )^2}-\frac {3 e^4}{\left (x+e^2\right )^2}+\frac {42}{\left (x+e^2\right )^2}-\frac {2 \left (1+e^2-e^4\right ) \log (x)}{e^4}-\frac {2 \left (7-2 e^2\right ) \log (x)}{e^4}+\frac {14 \log (x)}{e^4}+\frac {2 \left (1+e^2-e^4\right ) \log \left (25 e^x x\right )}{e^2 \left (x+e^2\right )}+\frac {2 \left (7-2 e^2\right ) \log \left (25 e^x x\right )}{\left (x+e^2\right )^2}-\frac {2 \left (7-8 e^6+9 e^8\right ) \log \left (x+e^2\right )}{e^4}+\frac {2 \left (1-2 e^4+e^6\right ) \log \left (x+e^2\right )}{e^4}+\frac {2 \left (7-2 e^2\right ) \log \left (x+e^2\right )}{e^4}+18 e^4 \log \left (x+e^2\right )-18 e^2 \log \left (x+e^2\right )+6 \log \left (x+e^2\right )\)

Input:

Int[(-84*x - 10*x^2 + 2*E^6*x^2 + 6*x^3 + 6*x^4 + 2*x^5 + E^4*(16*x + 10*x 
^2 + 6*x^3) + E^2*(14 + 46*x + 30*x^2 + 16*x^3 + 6*x^4) + (-26*x + 2*E^4*x 
 - 2*x^2 + E^2*(2 + 6*x + 2*x^2))*Log[25*E^x*x] - 2*x*Log[25*E^x*x]^2)/(E^ 
6*x + 3*E^4*x^2 + 3*E^2*x^3 + x^4),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(123\) vs. \(2(22)=44\).

Time = 1.39 (sec) , antiderivative size = 124, normalized size of antiderivative = 5.17

method	result	size
parallelrisch	\(-\frac {-49-28 x -2 \ln \left (25 \,{\mathrm e}^{x} x \right ) x^{2}-4 \ln \left (25 \,{\mathrm e}^{x} x \right ) x +2 x \,{\mathrm e}^{6}+8 x \,{\mathrm e}^{4}-14 \ln \left (25 \,{\mathrm e}^{x} x \right )+4 \,{\mathrm e}^{6}+{\mathrm e}^{8}-2 x^{3} {\mathrm e}^{2}+22 \,{\mathrm e}^{2} x +18 \,{\mathrm e}^{4}-4 x^{3}-x^{4}-\ln \left (25 \,{\mathrm e}^{x} x \right )^{2}-2 \,{\mathrm e}^{2} \ln \left (25 \,{\mathrm e}^{x} x \right ) x}{{\mathrm e}^{4}+2 \,{\mathrm e}^{2} x +x^{2}}\)	\(124\)
risch	\(\text {Expression too large to display}\)	\(941\)

Input:

int((-2*x*ln(25*exp(x)*x)^2+(2*x*exp(2)^2+(2*x^2+6*x+2)*exp(2)-2*x^2-26*x) 
*ln(25*exp(x)*x)+2*x^2*exp(2)^3+(6*x^3+10*x^2+16*x)*exp(2)^2+(6*x^4+16*x^3 
+30*x^2+46*x+14)*exp(2)+2*x^5+6*x^4+6*x^3-10*x^2-84*x)/(x*exp(2)^3+3*x^2*e 
xp(2)^2+3*x^3*exp(2)+x^4),x,method=_RETURNVERBOSE)
 

Output:

-(-49-28*x-2*ln(25*exp(x)*x)*x^2-4*ln(25*exp(x)*x)*x+2*x*exp(2)^3+8*x*exp( 
2)^2-14*ln(25*exp(x)*x)+4*exp(2)^3+exp(2)^4-2*x^3*exp(2)+22*exp(2)*x+18*ex 
p(2)^2-4*x^3-x^4-ln(25*exp(x)*x)^2-2*exp(2)*ln(25*exp(x)*x)*x)/(exp(2)^2+2 
*exp(2)*x+x^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (22) = 44\).

Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.58 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=\frac {x^{4} + 4 \, x^{3} + {\left (x^{2} + 8 \, x - 18\right )} e^{4} + 2 \, {\left (x^{3} + 4 \, x^{2} - 11 \, x\right )} e^{2} + 2 \, {\left (x^{2} + x e^{2} + 2 \, x + 7\right )} \log \left (25 \, x e^{x}\right ) + \log \left (25 \, x e^{x}\right )^{2} + 28 \, x + 4 \, e^{6} + 49}{x^{2} + 2 \, x e^{2} + e^{4}} \] Input:

integrate((-2*x*log(25*exp(x)*x)^2+(2*x*exp(2)^2+(2*x^2+6*x+2)*exp(2)-2*x^ 
2-26*x)*log(25*exp(x)*x)+2*x^2*exp(2)^3+(6*x^3+10*x^2+16*x)*exp(2)^2+(6*x^ 
4+16*x^3+30*x^2+46*x+14)*exp(2)+2*x^5+6*x^4+6*x^3-10*x^2-84*x)/(x*exp(2)^3 
+3*x^2*exp(2)^2+3*x^3*exp(2)+x^4),x, algorithm="fricas")
 

Output:

(x^4 + 4*x^3 + (x^2 + 8*x - 18)*e^4 + 2*(x^3 + 4*x^2 - 11*x)*e^2 + 2*(x^2 
+ x*e^2 + 2*x + 7)*log(25*x*e^x) + log(25*x*e^x)^2 + 28*x + 4*e^6 + 49)/(x 
^2 + 2*x*e^2 + e^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (20) = 40\).

Time = 0.78 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.67 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=x^{2} + 6 x + 2 \log {\left (x \right )} + \frac {x \left (- 22 e^{2} + 28 + 4 e^{4}\right ) - 18 e^{4} + 49 + 4 e^{6}}{x^{2} + 2 x e^{2} + e^{4}} + \frac {\left (- 2 x e^{2} + 4 x - 2 e^{4} + 14\right ) \log {\left (25 x e^{x} \right )}}{x^{2} + 2 x e^{2} + e^{4}} + \frac {\log {\left (25 x e^{x} \right )}^{2}}{x^{2} + 2 x e^{2} + e^{4}} \] Input:

integrate((-2*x*ln(25*exp(x)*x)**2+(2*x*exp(2)**2+(2*x**2+6*x+2)*exp(2)-2* 
x**2-26*x)*ln(25*exp(x)*x)+2*x**2*exp(2)**3+(6*x**3+10*x**2+16*x)*exp(2)** 
2+(6*x**4+16*x**3+30*x**2+46*x+14)*exp(2)+2*x**5+6*x**4+6*x**3-10*x**2-84* 
x)/(x*exp(2)**3+3*x**2*exp(2)**2+3*x**3*exp(2)+x**4),x)
 

Output:

x**2 + 6*x + 2*log(x) + (x*(-22*exp(2) + 28 + 4*exp(4)) - 18*exp(4) + 49 + 
 4*exp(6))/(x**2 + 2*x*exp(2) + exp(4)) + (-2*x*exp(2) + 4*x - 2*exp(4) + 
14)*log(25*x*exp(x))/(x**2 + 2*x*exp(2) + exp(4)) + log(25*x*exp(x))**2/(x 
**2 + 2*x*exp(2) + exp(4))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 548 vs. \(2 (22) = 44\).

Time = 0.19 (sec) , antiderivative size = 548, normalized size of antiderivative = 22.83 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx =\text {Too large to display} \] Input:

integrate((-2*x*log(25*exp(x)*x)^2+(2*x*exp(2)^2+(2*x^2+6*x+2)*exp(2)-2*x^ 
2-26*x)*log(25*exp(x)*x)+2*x^2*exp(2)^3+(6*x^3+10*x^2+16*x)*exp(2)^2+(6*x^ 
4+16*x^3+30*x^2+46*x+14)*exp(2)+2*x^5+6*x^4+6*x^3-10*x^2-84*x)/(x*exp(2)^3 
+3*x^2*exp(2)^2+3*x^3*exp(2)+x^4),x, algorithm="maxima")
 

Output:

2*(e^6 - 3*e^4 + 7)*e^(-4)*log(x + e^2) + x^2 + 3*((4*x*e^2 + 3*e^4)/(x^2 
+ 2*x*e^2 + e^4) + 2*log(x + e^2))*e^4 - 3*(6*e^2*log(x + e^2) - 2*x + (6* 
x*e^4 + 5*e^6)/(x^2 + 2*x*e^2 + e^4))*e^2 - 7*(2*e^(-6)*log(x + e^2) - 2*e 
^(-6)*log(x) - (2*x + 3*e^2)/(x^2*e^4 + 2*x*e^6 + e^8))*e^2 - 6*x*e^2 + 8* 
((4*x*e^2 + 3*e^4)/(x^2 + 2*x*e^2 + e^4) + 2*log(x + e^2))*e^2 + 12*e^4*lo 
g(x + e^2) - 18*e^2*log(x + e^2) + 6*x - (2*x + e^2)*e^6/(x^2 + 2*x*e^2 + 
e^4) - 5*(2*x + e^2)*e^4/(x^2 + 2*x*e^2 + e^4) - 15*(2*x + e^2)*e^2/(x^2 + 
 2*x*e^2 + e^4) + (e^4*log(x)^2 - 2*((2*log(5) + 7)*e^6 - 2*(3*log(5) + 8) 
*e^4 - e^8 + 7*e^2)*x - (4*log(5) + 9)*e^8 + 2*(2*log(5)^2 + 14*log(5) - 7 
)*e^4 + 2*(x^2*(e^4 - 7) + x*(e^6 + 3*e^4 - 14*e^2) + 2*e^4*log(5))*log(x) 
 + 2*e^10 + 18*e^6)/(x^2*e^4 + 2*x*e^6 + e^8) + (8*x*e^6 + 7*e^8)/(x^2 + 2 
*x*e^2 + e^4) - 3*(6*x*e^4 + 5*e^6)/(x^2 + 2*x*e^2 + e^4) + 3*(4*x*e^2 + 3 
*e^4)/(x^2 + 2*x*e^2 + e^4) + 5*(2*x + e^2)/(x^2 + 2*x*e^2 + e^4) - 8*e^4/ 
(x^2 + 2*x*e^2 + e^4) - 23*e^2/(x^2 + 2*x*e^2 + e^4) + 42/(x^2 + 2*x*e^2 + 
 e^4) + 6*log(x + e^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 126 vs. \(2 (22) = 44\).

Time = 0.13 (sec) , antiderivative size = 126, normalized size of antiderivative = 5.25 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=\frac {x^{4} + 2 \, x^{3} e^{2} + 6 \, x^{3} + x^{2} e^{4} + 12 \, x^{2} e^{2} - 4 \, x e^{2} \log \left (5\right ) + 2 \, x^{2} \log \left (x\right ) + 2 \, x e^{2} \log \left (x\right ) + 12 \, x e^{4} - 32 \, x e^{2} + 12 \, x \log \left (5\right ) - 4 \, e^{4} \log \left (5\right ) + 4 \, \log \left (5\right )^{2} + 6 \, x \log \left (x\right ) + 4 \, \log \left (5\right ) \log \left (x\right ) + \log \left (x\right )^{2} + 42 \, x + 6 \, e^{6} - 23 \, e^{4} + 28 \, \log \left (5\right ) + 14 \, \log \left (x\right ) + 49}{x^{2} + 2 \, x e^{2} + e^{4}} \] Input:

integrate((-2*x*log(25*exp(x)*x)^2+(2*x*exp(2)^2+(2*x^2+6*x+2)*exp(2)-2*x^ 
2-26*x)*log(25*exp(x)*x)+2*x^2*exp(2)^3+(6*x^3+10*x^2+16*x)*exp(2)^2+(6*x^ 
4+16*x^3+30*x^2+46*x+14)*exp(2)+2*x^5+6*x^4+6*x^3-10*x^2-84*x)/(x*exp(2)^3 
+3*x^2*exp(2)^2+3*x^3*exp(2)+x^4),x, algorithm="giac")
 

Output:

(x^4 + 2*x^3*e^2 + 6*x^3 + x^2*e^4 + 12*x^2*e^2 - 4*x*e^2*log(5) + 2*x^2*l 
og(x) + 2*x*e^2*log(x) + 12*x*e^4 - 32*x*e^2 + 12*x*log(5) - 4*e^4*log(5) 
+ 4*log(5)^2 + 6*x*log(x) + 4*log(5)*log(x) + log(x)^2 + 42*x + 6*e^6 - 23 
*e^4 + 28*log(5) + 14*log(x) + 49)/(x^2 + 2*x*e^2 + e^4)
 

Mupad [B] (verification not implemented)

Time = 3.63 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.71 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=6\,x+2\,\ln \left (x\right )+\frac {{\ln \left (25\,x\,{\mathrm {e}}^x\right )}^2}{x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4}+\frac {4\,{\mathrm {e}}^6-18\,{\mathrm {e}}^4+x\,\left (4\,{\mathrm {e}}^4-22\,{\mathrm {e}}^2+28\right )+49}{x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4}+x^2-\frac {\ln \left (25\,x\,{\mathrm {e}}^x\right )\,\left (2\,{\mathrm {e}}^2+{\mathrm {e}}^4+{\mathrm {e}}^2\,\left ({\mathrm {e}}^2-2\right )+x\,\left (2\,{\mathrm {e}}^2-4\right )-14\right )}{x^2+2\,{\mathrm {e}}^2\,x+{\mathrm {e}}^4} \] Input:

int((exp(4)*(16*x + 10*x^2 + 6*x^3) - 2*x*log(25*x*exp(x))^2 - log(25*x*ex 
p(x))*(26*x - exp(2)*(6*x + 2*x^2 + 2) - 2*x*exp(4) + 2*x^2) - 84*x + 2*x^ 
2*exp(6) + exp(2)*(46*x + 30*x^2 + 16*x^3 + 6*x^4 + 14) - 10*x^2 + 6*x^3 + 
 6*x^4 + 2*x^5)/(x*exp(6) + 3*x^3*exp(2) + 3*x^2*exp(4) + x^4),x)
 

Output:

6*x + 2*log(x) + log(25*x*exp(x))^2/(exp(4) + 2*x*exp(2) + x^2) + (4*exp(6 
) - 18*exp(4) + x*(4*exp(4) - 22*exp(2) + 28) + 49)/(exp(4) + 2*x*exp(2) + 
 x^2) + x^2 - (log(25*x*exp(x))*(2*exp(2) + exp(4) + exp(2)*(exp(2) - 2) + 
 x*(2*exp(2) - 4) - 14))/(exp(4) + 2*x*exp(2) + x^2)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 179, normalized size of antiderivative = 7.46 \[ \int \frac {-84 x-10 x^2+2 e^6 x^2+6 x^3+6 x^4+2 x^5+e^4 \left (16 x+10 x^2+6 x^3\right )+e^2 \left (14+46 x+30 x^2+16 x^3+6 x^4\right )+\left (-26 x+2 e^4 x-2 x^2+e^2 \left (2+6 x+2 x^2\right )\right ) \log \left (25 e^x x\right )-2 x \log ^2\left (25 e^x x\right )}{e^6 x+3 e^4 x^2+3 e^2 x^3+x^4} \, dx=\frac {\mathrm {log}\left (25 e^{x} x \right )^{2} e^{2}-2 \,\mathrm {log}\left (25 e^{x} x \right ) e^{6}-2 \,\mathrm {log}\left (25 e^{x} x \right ) e^{4} x +4 \,\mathrm {log}\left (25 e^{x} x \right ) e^{2} x +14 \,\mathrm {log}\left (25 e^{x} x \right ) e^{2}+2 \,\mathrm {log}\left (x \right ) e^{6}+4 \,\mathrm {log}\left (x \right ) e^{4} x +2 \,\mathrm {log}\left (x \right ) e^{2} x^{2}-e^{8}+e^{6} x^{2}-7 e^{6}+2 e^{4} x^{3}+7 e^{4} x^{2}-14 e^{4}+e^{2} x^{4}+6 e^{2} x^{3}+11 e^{2} x^{2}+49 e^{2}-14 x^{2}}{e^{2} \left (e^{4}+2 e^{2} x +x^{2}\right )} \] Input:

int((-2*x*log(25*exp(x)*x)^2+(2*x*exp(2)^2+(2*x^2+6*x+2)*exp(2)-2*x^2-26*x 
)*log(25*exp(x)*x)+2*x^2*exp(2)^3+(6*x^3+10*x^2+16*x)*exp(2)^2+(6*x^4+16*x 
^3+30*x^2+46*x+14)*exp(2)+2*x^5+6*x^4+6*x^3-10*x^2-84*x)/(x*exp(2)^3+3*x^2 
*exp(2)^2+3*x^3*exp(2)+x^4),x)
 

Output:

(log(25*e**x*x)**2*e**2 - 2*log(25*e**x*x)*e**6 - 2*log(25*e**x*x)*e**4*x 
+ 4*log(25*e**x*x)*e**2*x + 14*log(25*e**x*x)*e**2 + 2*log(x)*e**6 + 4*log 
(x)*e**4*x + 2*log(x)*e**2*x**2 - e**8 + e**6*x**2 - 7*e**6 + 2*e**4*x**3 
+ 7*e**4*x**2 - 14*e**4 + e**2*x**4 + 6*e**2*x**3 + 11*e**2*x**2 + 49*e**2 
 - 14*x**2)/(e**2*(e**4 + 2*e**2*x + x**2))