\(\int \frac {5 x^4+10 x^5+e^{4 x} (5+10 x)+e^{10} (42+90 x)+e^5 (31 x^2+60 x^3)+e^{2 x} (42+e^5 (-58-122 x)+90 x-10 x^2-20 x^3+e^{10} (5+10 x))+e^{3 x} (-29-61 x+e^5 (10+20 x))+e^x (e^{10} (-29-61 x)+31 x^2+59 x^3+e^5 (84+180 x-10 x^2-20 x^3))+(45 e^{10}+5 e^{4 x}+e^{3 x} (-30+10 e^5)+30 e^5 x^2+5 x^4+e^{2 x} (45-60 e^5+5 e^{10}-10 x^2)+e^x (-30 e^{10}+30 x^2+e^5 (90-10 x^2))) \log (2 x)}{9 e^{10}+e^{4 x}+e^{3 x} (-6+2 e^5)+6 e^5 x^2+x^4+e^{2 x} (9-12 e^5+e^{10}-2 x^2)+e^x (-6 e^{10}+6 x^2+e^5 (18-2 x^2))} \, dx\) [898]

Optimal result

Integrand size = 326, antiderivative size = 35 \[ \int \frac {5 x^4+10 x^5+e^{4 x} (5+10 x)+e^{10} (42+90 x)+e^5 \left (31 x^2+60 x^3\right )+e^{2 x} \left (42+e^5 (-58-122 x)+90 x-10 x^2-20 x^3+e^{10} (5+10 x)\right )+e^{3 x} \left (-29-61 x+e^5 (10+20 x)\right )+e^x \left (e^{10} (-29-61 x)+31 x^2+59 x^3+e^5 \left (84+180 x-10 x^2-20 x^3\right )\right )+\left (45 e^{10}+5 e^{4 x}+e^{3 x} \left (-30+10 e^5\right )+30 e^5 x^2+5 x^4+e^{2 x} \left (45-60 e^5+5 e^{10}-10 x^2\right )+e^x \left (-30 e^{10}+30 x^2+e^5 \left (90-10 x^2\right )\right )\right ) \log (2 x)}{9 e^{10}+e^{4 x}+e^{3 x} \left (-6+2 e^5\right )+6 e^5 x^2+x^4+e^{2 x} \left (9-12 e^5+e^{10}-2 x^2\right )+e^x \left (-6 e^{10}+6 x^2+e^5 \left (18-2 x^2\right )\right )} \, dx=-\frac {x}{3-e^x+\frac {x^2}{e^5+e^x}}+5 x (x+\log (2 x)) \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.43 \[ \int \frac {5 x^4+10 x^5+e^{4 x} (5+10 x)+e^{10} (42+90 x)+e^5 \left (31 x^2+60 x^3\right )+e^{2 x} \left (42+e^5 (-58-122 x)+90 x-10 x^2-20 x^3+e^{10} (5+10 x)\right )+e^{3 x} \left (-29-61 x+e^5 (10+20 x)\right )+e^x \left (e^{10} (-29-61 x)+31 x^2+59 x^3+e^5 \left (84+180 x-10 x^2-20 x^3\right )\right )+\left (45 e^{10}+5 e^{4 x}+e^{3 x} \left (-30+10 e^5\right )+30 e^5 x^2+5 x^4+e^{2 x} \left (45-60 e^5+5 e^{10}-10 x^2\right )+e^x \left (-30 e^{10}+30 x^2+e^5 \left (90-10 x^2\right )\right )\right ) \log (2 x)}{9 e^{10}+e^{4 x}+e^{3 x} \left (-6+2 e^5\right )+6 e^5 x^2+x^4+e^{2 x} \left (9-12 e^5+e^{10}-2 x^2\right )+e^x \left (-6 e^{10}+6 x^2+e^5 \left (18-2 x^2\right )\right )} \, dx=5 x^2+\frac {\left (e^5+e^x\right ) x}{-3 e^5-3 e^x+e^{2 x}+e^{5+x}-x^2}+5 x \log (2 x) \] Input:

Integrate[(5*x^4 + 10*x^5 + E^(4*x)*(5 + 10*x) + E^10*(42 + 90*x) + E^5*(3 
1*x^2 + 60*x^3) + E^(2*x)*(42 + E^5*(-58 - 122*x) + 90*x - 10*x^2 - 20*x^3 
 + E^10*(5 + 10*x)) + E^(3*x)*(-29 - 61*x + E^5*(10 + 20*x)) + E^x*(E^10*( 
-29 - 61*x) + 31*x^2 + 59*x^3 + E^5*(84 + 180*x - 10*x^2 - 20*x^3)) + (45* 
E^10 + 5*E^(4*x) + E^(3*x)*(-30 + 10*E^5) + 30*E^5*x^2 + 5*x^4 + E^(2*x)*( 
45 - 60*E^5 + 5*E^10 - 10*x^2) + E^x*(-30*E^10 + 30*x^2 + E^5*(90 - 10*x^2 
)))*Log[2*x])/(9*E^10 + E^(4*x) + E^(3*x)*(-6 + 2*E^5) + 6*E^5*x^2 + x^4 + 
 E^(2*x)*(9 - 12*E^5 + E^10 - 2*x^2) + E^x*(-6*E^10 + 6*x^2 + E^5*(18 - 2* 
x^2))),x]
 

Output:

5*x^2 + ((E^5 + E^x)*x)/(-3*E^5 - 3*E^x + E^(2*x) + E^(5 + x) - x^2) + 5*x 
*Log[2*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.

\(\displaystyle \int \frac {10 x^5+5 x^4+\left (5 x^4+30 e^5 x^2+e^{2 x} \left (-10 x^2+5 e^{10}-60 e^5+45\right )+e^x \left (30 x^2+e^5 \left (90-10 x^2\right )-30 e^{10}\right )+5 e^{4 x}+\left (10 e^5-30\right ) e^{3 x}+45 e^{10}\right ) \log (2 x)+e^5 \left (60 x^3+31 x^2\right )+e^{2 x} \left (-20 x^3-10 x^2+90 x+e^5 (-122 x-58)+e^{10} (10 x+5)+42\right )+e^x \left (59 x^3+31 x^2+e^5 \left (-20 x^3-10 x^2+180 x+84\right )+e^{10} (-61 x-29)\right )+e^{4 x} (10 x+5)+e^{10} (90 x+42)+e^{3 x} \left (-61 x+e^5 (20 x+10)-29\right )}{x^4+6 e^5 x^2+e^{2 x} \left (-2 x^2+e^{10}-12 e^5+9\right )+e^x \left (6 x^2+e^5 \left (18-2 x^2\right )-6 e^{10}\right )+e^{4 x}+\left (2 e^5-6\right ) e^{3 x}+9 e^{10}} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {10 x^5+5 x^4+\left (5 x^4+30 e^5 x^2+e^{2 x} \left (-10 x^2+5 e^{10}-60 e^5+45\right )+e^x \left (30 x^2+e^5 \left (90-10 x^2\right )-30 e^{10}\right )+5 e^{4 x}+\left (10 e^5-30\right ) e^{3 x}+45 e^{10}\right ) \log (2 x)+e^5 \left (60 x^3+31 x^2\right )+e^{2 x} \left (-20 x^3-10 x^2+90 x+e^5 (-122 x-58)+e^{10} (10 x+5)+42\right )+e^x \left (59 x^3+31 x^2+e^5 \left (-20 x^3-10 x^2+180 x+84\right )+e^{10} (-61 x-29)\right )+e^{4 x} (10 x+5)+e^{10} (90 x+42)+e^{3 x} \left (-61 x+e^5 (20 x+10)-29\right )}{\left (x^2-e^{2 x}+3 \left (1-\frac {e^5}{3}\right ) e^x+3 e^5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {e^x x+3 \left (1+\frac {e^5}{3}\right ) x-e^x-e^5}{x^2-e^{2 x}+3 \left (1-\frac {e^5}{3}\right ) e^x+3 e^5}+\frac {x \left (-2 e^x x^2-3 \left (1+\frac {e^5}{3}\right ) x^2+2 e^x x+2 e^5 x-9 \left (1+\frac {e^5}{3}\right ) e^x-9 e^5 \left (1+\frac {e^5}{3}\right )\right )}{\left (x^2-e^{2 x}+3 \left (1-\frac {e^5}{3}\right ) e^x+3 e^5\right )^2}+5 (2 x+\log (2 x)+1)\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

Int[(5*x^4 + 10*x^5 + E^(4*x)*(5 + 10*x) + E^10*(42 + 90*x) + E^5*(31*x^2 
+ 60*x^3) + E^(2*x)*(42 + E^5*(-58 - 122*x) + 90*x - 10*x^2 - 20*x^3 + E^1 
0*(5 + 10*x)) + E^(3*x)*(-29 - 61*x + E^5*(10 + 20*x)) + E^x*(E^10*(-29 - 
61*x) + 31*x^2 + 59*x^3 + E^5*(84 + 180*x - 10*x^2 - 20*x^3)) + (45*E^10 + 
 5*E^(4*x) + E^(3*x)*(-30 + 10*E^5) + 30*E^5*x^2 + 5*x^4 + E^(2*x)*(45 - 6 
0*E^5 + 5*E^10 - 10*x^2) + E^x*(-30*E^10 + 30*x^2 + E^5*(90 - 10*x^2)))*Lo 
g[2*x])/(9*E^10 + E^(4*x) + E^(3*x)*(-6 + 2*E^5) + 6*E^5*x^2 + x^4 + E^(2* 
x)*(9 - 12*E^5 + E^10 - 2*x^2) + E^x*(-6*E^10 + 6*x^2 + E^5*(18 - 2*x^2))) 
,x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(68\) vs. \(2(32)=64\).

Time = 0.13 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.97

\[5 x \ln \left (2 x \right )+\frac {x \left (5 x \,{\mathrm e}^{5+x}-5 x^{3}+5 x \,{\mathrm e}^{2 x}-15 x \,{\mathrm e}^{5}-15 \,{\mathrm e}^{x} x +{\mathrm e}^{5}+{\mathrm e}^{x}\right )}{{\mathrm e}^{5+x}+{\mathrm e}^{2 x}-x^{2}-3 \,{\mathrm e}^{5}-3 \,{\mathrm e}^{x}}\]

Input:

int(((5*exp(x)^4+(10*exp(5)-30)*exp(x)^3+(5*exp(5)^2-60*exp(5)-10*x^2+45)* 
exp(x)^2+(-30*exp(5)^2+(-10*x^2+90)*exp(5)+30*x^2)*exp(x)+45*exp(5)^2+30*x 
^2*exp(5)+5*x^4)*ln(2*x)+(10*x+5)*exp(x)^4+((20*x+10)*exp(5)-61*x-29)*exp( 
x)^3+((10*x+5)*exp(5)^2+(-122*x-58)*exp(5)-20*x^3-10*x^2+90*x+42)*exp(x)^2 
+((-61*x-29)*exp(5)^2+(-20*x^3-10*x^2+180*x+84)*exp(5)+59*x^3+31*x^2)*exp( 
x)+(90*x+42)*exp(5)^2+(60*x^3+31*x^2)*exp(5)+10*x^5+5*x^4)/(exp(x)^4+(2*ex 
p(5)-6)*exp(x)^3+(exp(5)^2-12*exp(5)-2*x^2+9)*exp(x)^2+(-6*exp(5)^2+(-2*x^ 
2+18)*exp(5)+6*x^2)*exp(x)+9*exp(5)^2+6*x^2*exp(5)+x^4),x)
 

Output:

5*x*ln(2*x)+x*(5*x*exp(5+x)-5*x^3+5*x*exp(2*x)-15*x*exp(5)-15*exp(x)*x+exp 
(5)+exp(x))/(exp(5+x)+exp(2*x)-x^2-3*exp(5)-3*exp(x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (32) = 64\).

Time = 0.10 (sec) , antiderivative size = 104, normalized size of antiderivative = 2.97 \[ \int \frac {5 x^4+10 x^5+e^{4 x} (5+10 x)+e^{10} (42+90 x)+e^5 \left (31 x^2+60 x^3\right )+e^{2 x} \left (42+e^5 (-58-122 x)+90 x-10 x^2-20 x^3+e^{10} (5+10 x)\right )+e^{3 x} \left (-29-61 x+e^5 (10+20 x)\right )+e^x \left (e^{10} (-29-61 x)+31 x^2+59 x^3+e^5 \left (84+180 x-10 x^2-20 x^3\right )\right )+\left (45 e^{10}+5 e^{4 x}+e^{3 x} \left (-30+10 e^5\right )+30 e^5 x^2+5 x^4+e^{2 x} \left (45-60 e^5+5 e^{10}-10 x^2\right )+e^x \left (-30 e^{10}+30 x^2+e^5 \left (90-10 x^2\right )\right )\right ) \log (2 x)}{9 e^{10}+e^{4 x}+e^{3 x} \left (-6+2 e^5\right )+6 e^5 x^2+x^4+e^{2 x} \left (9-12 e^5+e^{10}-2 x^2\right )+e^x \left (-6 e^{10}+6 x^2+e^5 \left (18-2 x^2\right )\right )} \, dx=\frac {5 \, x^{4} - 5 \, x^{2} e^{\left (2 \, x\right )} + {\left (15 \, x^{2} - x\right )} e^{5} - {\left (5 \, x^{2} e^{5} - 15 \, x^{2} + x\right )} e^{x} + 5 \, {\left (x^{3} + 3 \, x e^{5} - x e^{\left (2 \, x\right )} - {\left (x e^{5} - 3 \, x\right )} e^{x}\right )} \log \left (2 \, x\right )}{x^{2} - {\left (e^{5} - 3\right )} e^{x} + 3 \, e^{5} - e^{\left (2 \, x\right )}} \] Input:

integrate(((5*exp(x)^4+(10*exp(5)-30)*exp(x)^3+(5*exp(5)^2-60*exp(5)-10*x^ 
2+45)*exp(x)^2+(-30*exp(5)^2+(-10*x^2+90)*exp(5)+30*x^2)*exp(x)+45*exp(5)^ 
2+30*x^2*exp(5)+5*x^4)*log(2*x)+(10*x+5)*exp(x)^4+((20*x+10)*exp(5)-61*x-2 
9)*exp(x)^3+((10*x+5)*exp(5)^2+(-122*x-58)*exp(5)-20*x^3-10*x^2+90*x+42)*e 
xp(x)^2+((-61*x-29)*exp(5)^2+(-20*x^3-10*x^2+180*x+84)*exp(5)+59*x^3+31*x^ 
2)*exp(x)+(90*x+42)*exp(5)^2+(60*x^3+31*x^2)*exp(5)+10*x^5+5*x^4)/(exp(x)^ 
4+(2*exp(5)-6)*exp(x)^3+(exp(5)^2-12*exp(5)-2*x^2+9)*exp(x)^2+(-6*exp(5)^2 
+(-2*x^2+18)*exp(5)+6*x^2)*exp(x)+9*exp(5)^2+6*x^2*exp(5)+x^4),x, algorith 
m="fricas")
 

Output:

(5*x^4 - 5*x^2*e^(2*x) + (15*x^2 - x)*e^5 - (5*x^2*e^5 - 15*x^2 + x)*e^x + 
 5*(x^3 + 3*x*e^5 - x*e^(2*x) - (x*e^5 - 3*x)*e^x)*log(2*x))/(x^2 - (e^5 - 
 3)*e^x + 3*e^5 - e^(2*x))
 

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.26 \[ \int \frac {5 x^4+10 x^5+e^{4 x} (5+10 x)+e^{10} (42+90 x)+e^5 \left (31 x^2+60 x^3\right )+e^{2 x} \left (42+e^5 (-58-122 x)+90 x-10 x^2-20 x^3+e^{10} (5+10 x)\right )+e^{3 x} \left (-29-61 x+e^5 (10+20 x)\right )+e^x \left (e^{10} (-29-61 x)+31 x^2+59 x^3+e^5 \left (84+180 x-10 x^2-20 x^3\right )\right )+\left (45 e^{10}+5 e^{4 x}+e^{3 x} \left (-30+10 e^5\right )+30 e^5 x^2+5 x^4+e^{2 x} \left (45-60 e^5+5 e^{10}-10 x^2\right )+e^x \left (-30 e^{10}+30 x^2+e^5 \left (90-10 x^2\right )\right )\right ) \log (2 x)}{9 e^{10}+e^{4 x}+e^{3 x} \left (-6+2 e^5\right )+6 e^5 x^2+x^4+e^{2 x} \left (9-12 e^5+e^{10}-2 x^2\right )+e^x \left (-6 e^{10}+6 x^2+e^5 \left (18-2 x^2\right )\right )} \, dx=5 x^{2} + 5 x \log {\left (2 x \right )} + \frac {x e^{x} + x e^{5}}{- x^{2} + e^{2 x} + \left (-3 + e^{5}\right ) e^{x} - 3 e^{5}} \] Input:

integrate(((5*exp(x)**4+(10*exp(5)-30)*exp(x)**3+(5*exp(5)**2-60*exp(5)-10 
*x**2+45)*exp(x)**2+(-30*exp(5)**2+(-10*x**2+90)*exp(5)+30*x**2)*exp(x)+45 
*exp(5)**2+30*x**2*exp(5)+5*x**4)*ln(2*x)+(10*x+5)*exp(x)**4+((20*x+10)*ex 
p(5)-61*x-29)*exp(x)**3+((10*x+5)*exp(5)**2+(-122*x-58)*exp(5)-20*x**3-10* 
x**2+90*x+42)*exp(x)**2+((-61*x-29)*exp(5)**2+(-20*x**3-10*x**2+180*x+84)* 
exp(5)+59*x**3+31*x**2)*exp(x)+(90*x+42)*exp(5)**2+(60*x**3+31*x**2)*exp(5 
)+10*x**5+5*x**4)/(exp(x)**4+(2*exp(5)-6)*exp(x)**3+(exp(5)**2-12*exp(5)-2 
*x**2+9)*exp(x)**2+(-6*exp(5)**2+(-2*x**2+18)*exp(5)+6*x**2)*exp(x)+9*exp( 
5)**2+6*x**2*exp(5)+x**4),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 123 vs. \(2 (32) = 64\).

Time = 0.25 (sec) , antiderivative size = 123, normalized size of antiderivative = 3.51 \[ \int \frac {5 x^4+10 x^5+e^{4 x} (5+10 x)+e^{10} (42+90 x)+e^5 \left (31 x^2+60 x^3\right )+e^{2 x} \left (42+e^5 (-58-122 x)+90 x-10 x^2-20 x^3+e^{10} (5+10 x)\right )+e^{3 x} \left (-29-61 x+e^5 (10+20 x)\right )+e^x \left (e^{10} (-29-61 x)+31 x^2+59 x^3+e^5 \left (84+180 x-10 x^2-20 x^3\right )\right )+\left (45 e^{10}+5 e^{4 x}+e^{3 x} \left (-30+10 e^5\right )+30 e^5 x^2+5 x^4+e^{2 x} \left (45-60 e^5+5 e^{10}-10 x^2\right )+e^x \left (-30 e^{10}+30 x^2+e^5 \left (90-10 x^2\right )\right )\right ) \log (2 x)}{9 e^{10}+e^{4 x}+e^{3 x} \left (-6+2 e^5\right )+6 e^5 x^2+x^4+e^{2 x} \left (9-12 e^5+e^{10}-2 x^2\right )+e^x \left (-6 e^{10}+6 x^2+e^5 \left (18-2 x^2\right )\right )} \, dx=\frac {5 \, x^{4} + 5 \, x^{3} \log \left (2\right ) + 15 \, x^{2} e^{5} + x {\left (15 \, \log \left (2\right ) - 1\right )} e^{5} - 5 \, {\left (x^{2} + x \log \left (2\right ) + x \log \left (x\right )\right )} e^{\left (2 \, x\right )} - {\left (5 \, x^{2} {\left (e^{5} - 3\right )} + 5 \, x {\left (e^{5} - 3\right )} \log \left (x\right ) + {\left (5 \, e^{5} \log \left (2\right ) - 15 \, \log \left (2\right ) + 1\right )} x\right )} e^{x} + 5 \, {\left (x^{3} + 3 \, x e^{5}\right )} \log \left (x\right )}{x^{2} - {\left (e^{5} - 3\right )} e^{x} + 3 \, e^{5} - e^{\left (2 \, x\right )}} \] Input:

integrate(((5*exp(x)^4+(10*exp(5)-30)*exp(x)^3+(5*exp(5)^2-60*exp(5)-10*x^ 
2+45)*exp(x)^2+(-30*exp(5)^2+(-10*x^2+90)*exp(5)+30*x^2)*exp(x)+45*exp(5)^ 
2+30*x^2*exp(5)+5*x^4)*log(2*x)+(10*x+5)*exp(x)^4+((20*x+10)*exp(5)-61*x-2 
9)*exp(x)^3+((10*x+5)*exp(5)^2+(-122*x-58)*exp(5)-20*x^3-10*x^2+90*x+42)*e 
xp(x)^2+((-61*x-29)*exp(5)^2+(-20*x^3-10*x^2+180*x+84)*exp(5)+59*x^3+31*x^ 
2)*exp(x)+(90*x+42)*exp(5)^2+(60*x^3+31*x^2)*exp(5)+10*x^5+5*x^4)/(exp(x)^ 
4+(2*exp(5)-6)*exp(x)^3+(exp(5)^2-12*exp(5)-2*x^2+9)*exp(x)^2+(-6*exp(5)^2 
+(-2*x^2+18)*exp(5)+6*x^2)*exp(x)+9*exp(5)^2+6*x^2*exp(5)+x^4),x, algorith 
m="maxima")
 

Output:

(5*x^4 + 5*x^3*log(2) + 15*x^2*e^5 + x*(15*log(2) - 1)*e^5 - 5*(x^2 + x*lo 
g(2) + x*log(x))*e^(2*x) - (5*x^2*(e^5 - 3) + 5*x*(e^5 - 3)*log(x) + (5*e^ 
5*log(2) - 15*log(2) + 1)*x)*e^x + 5*(x^3 + 3*x*e^5)*log(x))/(x^2 - (e^5 - 
 3)*e^x + 3*e^5 - e^(2*x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1618 vs. \(2 (32) = 64\).

Time = 1.22 (sec) , antiderivative size = 1618, normalized size of antiderivative = 46.23 \[ \int \frac {5 x^4+10 x^5+e^{4 x} (5+10 x)+e^{10} (42+90 x)+e^5 \left (31 x^2+60 x^3\right )+e^{2 x} \left (42+e^5 (-58-122 x)+90 x-10 x^2-20 x^3+e^{10} (5+10 x)\right )+e^{3 x} \left (-29-61 x+e^5 (10+20 x)\right )+e^x \left (e^{10} (-29-61 x)+31 x^2+59 x^3+e^5 \left (84+180 x-10 x^2-20 x^3\right )\right )+\left (45 e^{10}+5 e^{4 x}+e^{3 x} \left (-30+10 e^5\right )+30 e^5 x^2+5 x^4+e^{2 x} \left (45-60 e^5+5 e^{10}-10 x^2\right )+e^x \left (-30 e^{10}+30 x^2+e^5 \left (90-10 x^2\right )\right )\right ) \log (2 x)}{9 e^{10}+e^{4 x}+e^{3 x} \left (-6+2 e^5\right )+6 e^5 x^2+x^4+e^{2 x} \left (9-12 e^5+e^{10}-2 x^2\right )+e^x \left (-6 e^{10}+6 x^2+e^5 \left (18-2 x^2\right )\right )} \, dx=\text {Too large to display} \] Input:

integrate(((5*exp(x)^4+(10*exp(5)-30)*exp(x)^3+(5*exp(5)^2-60*exp(5)-10*x^ 
2+45)*exp(x)^2+(-30*exp(5)^2+(-10*x^2+90)*exp(5)+30*x^2)*exp(x)+45*exp(5)^ 
2+30*x^2*exp(5)+5*x^4)*log(2*x)+(10*x+5)*exp(x)^4+((20*x+10)*exp(5)-61*x-2 
9)*exp(x)^3+((10*x+5)*exp(5)^2+(-122*x-58)*exp(5)-20*x^3-10*x^2+90*x+42)*e 
xp(x)^2+((-61*x-29)*exp(5)^2+(-20*x^3-10*x^2+180*x+84)*exp(5)+59*x^3+31*x^ 
2)*exp(x)+(90*x+42)*exp(5)^2+(60*x^3+31*x^2)*exp(5)+10*x^5+5*x^4)/(exp(x)^ 
4+(2*exp(5)-6)*exp(x)^3+(exp(5)^2-12*exp(5)-2*x^2+9)*exp(x)^2+(-6*exp(5)^2 
+(-2*x^2+18)*exp(5)+6*x^2)*exp(x)+9*exp(5)^2+6*x^2*exp(5)+x^4),x, algorith 
m="giac")
 

Output:

(20*x^8 + 20*x^7*log(2) + 20*x^7*log(x) - 20*x^7 + 5*x^6*e^10 + 150*x^6*e^ 
5 - 20*x^6*e^(2*x) - 20*x^6*e^(x + 5) + 60*x^6*e^x - 30*x^6*log(2) + 5*x^5 
*e^10*log(2) + 150*x^5*e^5*log(2) - 20*x^5*e^(2*x)*log(2) - 20*x^5*e^(x + 
5)*log(2) + 60*x^5*e^x*log(2) - 30*x^6*log(x) + 5*x^5*e^10*log(x) + 150*x^ 
5*e^5*log(x) - 20*x^5*e^(2*x)*log(x) - 20*x^5*e^(x + 5)*log(x) + 60*x^5*e^ 
x*log(x) + 55*x^6 - 64*x^5*e^5 + 40*x^5*e^(2*x) + 50*x^5*e^(x + 5) - 154*x 
^5*e^x + 55*x^5*log(2) - 5*x^4*e^10*log(2) - 120*x^4*e^5*log(2) + 40*x^4*e 
^(2*x)*log(2) + 45*x^4*e^(x + 5)*log(2) - 135*x^4*e^x*log(2) + 55*x^5*log( 
x) - 5*x^4*e^10*log(x) - 120*x^4*e^5*log(x) + 40*x^4*e^(2*x)*log(x) + 45*x 
^4*e^(x + 5)*log(x) - 135*x^4*e^x*log(x) - 10*x^5 + 30*x^4*e^15 + 365*x^4* 
e^10 + 280*x^4*e^5 - 65*x^4*e^(2*x) - 5*x^4*e^(2*x + 10) - 90*x^4*e^(2*x + 
 5) - 5*x^4*e^(x + 15) - 75*x^4*e^(x + 10) + 210*x^4*e^(x + 5) + 188*x^4*e 
^x - 45*x^4*log(2) + 30*x^3*e^15*log(2) + 360*x^3*e^10*log(2) + 270*x^3*e^ 
5*log(2) - 65*x^3*e^(2*x)*log(2) - 5*x^3*e^(2*x + 10)*log(2) - 90*x^3*e^(2 
*x + 5)*log(2) - 5*x^3*e^(x + 15)*log(2) - 75*x^3*e^(x + 10)*log(2) + 205* 
x^3*e^(x + 5)*log(2) + 195*x^3*e^x*log(2) - 45*x^4*log(x) + 30*x^3*e^15*lo 
g(x) + 360*x^3*e^10*log(x) + 270*x^3*e^5*log(x) - 65*x^3*e^(2*x)*log(x) - 
5*x^3*e^(2*x + 10)*log(x) - 90*x^3*e^(2*x + 5)*log(x) - 5*x^3*e^(x + 15)*l 
og(x) - 75*x^3*e^(x + 10)*log(x) + 205*x^3*e^(x + 5)*log(x) + 195*x^3*e^x* 
log(x) + 45*x^4 + 29*x^3*e^15 - 18*x^3*e^10 + 195*x^3*e^5 + 90*x^3*e^(2...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {5 x^4+10 x^5+e^{4 x} (5+10 x)+e^{10} (42+90 x)+e^5 \left (31 x^2+60 x^3\right )+e^{2 x} \left (42+e^5 (-58-122 x)+90 x-10 x^2-20 x^3+e^{10} (5+10 x)\right )+e^{3 x} \left (-29-61 x+e^5 (10+20 x)\right )+e^x \left (e^{10} (-29-61 x)+31 x^2+59 x^3+e^5 \left (84+180 x-10 x^2-20 x^3\right )\right )+\left (45 e^{10}+5 e^{4 x}+e^{3 x} \left (-30+10 e^5\right )+30 e^5 x^2+5 x^4+e^{2 x} \left (45-60 e^5+5 e^{10}-10 x^2\right )+e^x \left (-30 e^{10}+30 x^2+e^5 \left (90-10 x^2\right )\right )\right ) \log (2 x)}{9 e^{10}+e^{4 x}+e^{3 x} \left (-6+2 e^5\right )+6 e^5 x^2+x^4+e^{2 x} \left (9-12 e^5+e^{10}-2 x^2\right )+e^x \left (-6 e^{10}+6 x^2+e^5 \left (18-2 x^2\right )\right )} \, dx=\int \frac {{\mathrm {e}}^{2\,x}\,\left (90\,x-10\,x^2-20\,x^3+{\mathrm {e}}^{10}\,\left (10\,x+5\right )-{\mathrm {e}}^5\,\left (122\,x+58\right )+42\right )+{\mathrm {e}}^x\,\left ({\mathrm {e}}^5\,\left (-20\,x^3-10\,x^2+180\,x+84\right )+31\,x^2+59\,x^3-{\mathrm {e}}^{10}\,\left (61\,x+29\right )\right )+{\mathrm {e}}^5\,\left (60\,x^3+31\,x^2\right )+\ln \left (2\,x\right )\,\left (5\,{\mathrm {e}}^{4\,x}+45\,{\mathrm {e}}^{10}+30\,x^2\,{\mathrm {e}}^5-{\mathrm {e}}^{2\,x}\,\left (10\,x^2+60\,{\mathrm {e}}^5-5\,{\mathrm {e}}^{10}-45\right )+{\mathrm {e}}^{3\,x}\,\left (10\,{\mathrm {e}}^5-30\right )+5\,x^4-{\mathrm {e}}^x\,\left (30\,{\mathrm {e}}^{10}+{\mathrm {e}}^5\,\left (10\,x^2-90\right )-30\,x^2\right )\right )-{\mathrm {e}}^{3\,x}\,\left (61\,x-{\mathrm {e}}^5\,\left (20\,x+10\right )+29\right )+{\mathrm {e}}^{4\,x}\,\left (10\,x+5\right )+5\,x^4+10\,x^5+{\mathrm {e}}^{10}\,\left (90\,x+42\right )}{{\mathrm {e}}^{4\,x}+9\,{\mathrm {e}}^{10}+6\,x^2\,{\mathrm {e}}^5-{\mathrm {e}}^{2\,x}\,\left (2\,x^2+12\,{\mathrm {e}}^5-{\mathrm {e}}^{10}-9\right )+{\mathrm {e}}^{3\,x}\,\left (2\,{\mathrm {e}}^5-6\right )+x^4-{\mathrm {e}}^x\,\left (6\,{\mathrm {e}}^{10}+{\mathrm {e}}^5\,\left (2\,x^2-18\right )-6\,x^2\right )} \,d x \] Input:

int((exp(2*x)*(90*x - 10*x^2 - 20*x^3 + exp(10)*(10*x + 5) - exp(5)*(122*x 
 + 58) + 42) + exp(x)*(exp(5)*(180*x - 10*x^2 - 20*x^3 + 84) + 31*x^2 + 59 
*x^3 - exp(10)*(61*x + 29)) + exp(5)*(31*x^2 + 60*x^3) + log(2*x)*(5*exp(4 
*x) + 45*exp(10) + 30*x^2*exp(5) - exp(2*x)*(60*exp(5) - 5*exp(10) + 10*x^ 
2 - 45) + exp(3*x)*(10*exp(5) - 30) + 5*x^4 - exp(x)*(30*exp(10) + exp(5)* 
(10*x^2 - 90) - 30*x^2)) - exp(3*x)*(61*x - exp(5)*(20*x + 10) + 29) + exp 
(4*x)*(10*x + 5) + 5*x^4 + 10*x^5 + exp(10)*(90*x + 42))/(exp(4*x) + 9*exp 
(10) + 6*x^2*exp(5) - exp(2*x)*(12*exp(5) - exp(10) + 2*x^2 - 9) + exp(3*x 
)*(2*exp(5) - 6) + x^4 - exp(x)*(6*exp(10) + exp(5)*(2*x^2 - 18) - 6*x^2)) 
,x)
 

Output:

int((exp(2*x)*(90*x - 10*x^2 - 20*x^3 + exp(10)*(10*x + 5) - exp(5)*(122*x 
 + 58) + 42) + exp(x)*(exp(5)*(180*x - 10*x^2 - 20*x^3 + 84) + 31*x^2 + 59 
*x^3 - exp(10)*(61*x + 29)) + exp(5)*(31*x^2 + 60*x^3) + log(2*x)*(5*exp(4 
*x) + 45*exp(10) + 30*x^2*exp(5) - exp(2*x)*(60*exp(5) - 5*exp(10) + 10*x^ 
2 - 45) + exp(3*x)*(10*exp(5) - 30) + 5*x^4 - exp(x)*(30*exp(10) + exp(5)* 
(10*x^2 - 90) - 30*x^2)) - exp(3*x)*(61*x - exp(5)*(20*x + 10) + 29) + exp 
(4*x)*(10*x + 5) + 5*x^4 + 10*x^5 + exp(10)*(90*x + 42))/(exp(4*x) + 9*exp 
(10) + 6*x^2*exp(5) - exp(2*x)*(12*exp(5) - exp(10) + 2*x^2 - 9) + exp(3*x 
)*(2*exp(5) - 6) + x^4 - exp(x)*(6*exp(10) + exp(5)*(2*x^2 - 18) - 6*x^2)) 
, x)
 

Reduce [F]

\[ \int \frac {5 x^4+10 x^5+e^{4 x} (5+10 x)+e^{10} (42+90 x)+e^5 \left (31 x^2+60 x^3\right )+e^{2 x} \left (42+e^5 (-58-122 x)+90 x-10 x^2-20 x^3+e^{10} (5+10 x)\right )+e^{3 x} \left (-29-61 x+e^5 (10+20 x)\right )+e^x \left (e^{10} (-29-61 x)+31 x^2+59 x^3+e^5 \left (84+180 x-10 x^2-20 x^3\right )\right )+\left (45 e^{10}+5 e^{4 x}+e^{3 x} \left (-30+10 e^5\right )+30 e^5 x^2+5 x^4+e^{2 x} \left (45-60 e^5+5 e^{10}-10 x^2\right )+e^x \left (-30 e^{10}+30 x^2+e^5 \left (90-10 x^2\right )\right )\right ) \log (2 x)}{9 e^{10}+e^{4 x}+e^{3 x} \left (-6+2 e^5\right )+6 e^5 x^2+x^4+e^{2 x} \left (9-12 e^5+e^{10}-2 x^2\right )+e^x \left (-6 e^{10}+6 x^2+e^5 \left (18-2 x^2\right )\right )} \, dx=\text {too large to display} \] Input:

int(((5*exp(x)^4+(10*exp(5)-30)*exp(x)^3+(5*exp(5)^2-60*exp(5)-10*x^2+45)* 
exp(x)^2+(-30*exp(5)^2+(-10*x^2+90)*exp(5)+30*x^2)*exp(x)+45*exp(5)^2+30*x 
^2*exp(5)+5*x^4)*log(2*x)+(10*x+5)*exp(x)^4+((20*x+10)*exp(5)-61*x-29)*exp 
(x)^3+((10*x+5)*exp(5)^2+(-122*x-58)*exp(5)-20*x^3-10*x^2+90*x+42)*exp(x)^ 
2+((-61*x-29)*exp(5)^2+(-20*x^3-10*x^2+180*x+84)*exp(5)+59*x^3+31*x^2)*exp 
(x)+(90*x+42)*exp(5)^2+(60*x^3+31*x^2)*exp(5)+10*x^5+5*x^4)/(exp(x)^4+(2*e 
xp(5)-6)*exp(x)^3+(exp(5)^2-12*exp(5)-2*x^2+9)*exp(x)^2+(-6*exp(5)^2+(-2*x 
^2+18)*exp(5)+6*x^2)*exp(x)+9*exp(5)^2+6*x^2*exp(5)+x^4),x)
 

Output:

5*int(e**(4*x)/(e**(4*x) + 2*e**(3*x)*e**5 - 6*e**(3*x) + e**(2*x)*e**10 - 
 12*e**(2*x)*e**5 - 2*e**(2*x)*x**2 + 9*e**(2*x) - 6*e**x*e**10 - 2*e**x*e 
**5*x**2 + 18*e**x*e**5 + 6*e**x*x**2 + 9*e**10 + 6*e**5*x**2 + x**4),x) + 
 10*int(e**(3*x)/(e**(4*x) + 2*e**(3*x)*e**5 - 6*e**(3*x) + e**(2*x)*e**10 
 - 12*e**(2*x)*e**5 - 2*e**(2*x)*x**2 + 9*e**(2*x) - 6*e**x*e**10 - 2*e**x 
*e**5*x**2 + 18*e**x*e**5 + 6*e**x*x**2 + 9*e**10 + 6*e**5*x**2 + x**4),x) 
*e**5 - 29*int(e**(3*x)/(e**(4*x) + 2*e**(3*x)*e**5 - 6*e**(3*x) + e**(2*x 
)*e**10 - 12*e**(2*x)*e**5 - 2*e**(2*x)*x**2 + 9*e**(2*x) - 6*e**x*e**10 - 
 2*e**x*e**5*x**2 + 18*e**x*e**5 + 6*e**x*x**2 + 9*e**10 + 6*e**5*x**2 + x 
**4),x) + 5*int(e**(2*x)/(e**(4*x) + 2*e**(3*x)*e**5 - 6*e**(3*x) + e**(2* 
x)*e**10 - 12*e**(2*x)*e**5 - 2*e**(2*x)*x**2 + 9*e**(2*x) - 6*e**x*e**10 
- 2*e**x*e**5*x**2 + 18*e**x*e**5 + 6*e**x*x**2 + 9*e**10 + 6*e**5*x**2 + 
x**4),x)*e**10 - 58*int(e**(2*x)/(e**(4*x) + 2*e**(3*x)*e**5 - 6*e**(3*x) 
+ e**(2*x)*e**10 - 12*e**(2*x)*e**5 - 2*e**(2*x)*x**2 + 9*e**(2*x) - 6*e** 
x*e**10 - 2*e**x*e**5*x**2 + 18*e**x*e**5 + 6*e**x*x**2 + 9*e**10 + 6*e**5 
*x**2 + x**4),x)*e**5 + 42*int(e**(2*x)/(e**(4*x) + 2*e**(3*x)*e**5 - 6*e* 
*(3*x) + e**(2*x)*e**10 - 12*e**(2*x)*e**5 - 2*e**(2*x)*x**2 + 9*e**(2*x) 
- 6*e**x*e**10 - 2*e**x*e**5*x**2 + 18*e**x*e**5 + 6*e**x*x**2 + 9*e**10 + 
 6*e**5*x**2 + x**4),x) - 29*int(e**x/(e**(4*x) + 2*e**(3*x)*e**5 - 6*e**( 
3*x) + e**(2*x)*e**10 - 12*e**(2*x)*e**5 - 2*e**(2*x)*x**2 + 9*e**(2*x)...