\(\int \frac {125 e^2-125 x-250 e x+125 x^2+e^{6 x} (-e^2+x+2 e x-x^2)+e^{5 x} (3 e^2 x-3 x^2-6 e x^2+3 x^3)+e^x (75 e^2 x-75 x^2-150 e x^2+75 x^3)+e^{2 x} (10+75 x-75 x^2-15 x^3+15 x^4+e^2 (-75+15 x^2)+e (150 x-30 x^3))+e^{4 x} (-2-15 x+15 x^2+3 x^3-3 x^4+e^2 (15-3 x^2)+e (-30 x+6 x^3))+e^{3 x} (2 x+30 x^2-30 x^3-x^4+x^5+e^2 (-30 x+x^3)+e (60 x^2-2 x^4))+(-250 e x+250 x^2+e^{6 x} (2 e x-2 x^2)+e^{5 x} (-6 e x^2+6 x^3)+e^x (-150 e x^2+150 x^3)+e^{2 x} (20 x-150 x^2+30 x^4+e (150 x-30 x^3))+e^{4 x} (4 x+30 x^2-6 x^4+e (-30 x+6 x^3))+e^{3 x} (-4 x-60 x^3+2 x^5+e (60 x^2-2 x^4))) \log (x)}{-125 x+e^{6 x} x-75 e^x x^2-3 e^{5 x} x^2+e^{2 x} (75 x-15 x^3)+e^{4 x} (-15 x+3 x^3)+e^{3 x} (30 x^2-x^4)} \, dx\) [101]

Optimal result

Integrand size = 481, antiderivative size = 32 \[ \int \frac {125 e^2-125 x-250 e x+125 x^2+e^{6 x} \left (-e^2+x+2 e x-x^2\right )+e^{5 x} \left (3 e^2 x-3 x^2-6 e x^2+3 x^3\right )+e^x \left (75 e^2 x-75 x^2-150 e x^2+75 x^3\right )+e^{2 x} \left (10+75 x-75 x^2-15 x^3+15 x^4+e^2 \left (-75+15 x^2\right )+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (-2-15 x+15 x^2+3 x^3-3 x^4+e^2 \left (15-3 x^2\right )+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (2 x+30 x^2-30 x^3-x^4+x^5+e^2 \left (-30 x+x^3\right )+e \left (60 x^2-2 x^4\right )\right )+\left (-250 e x+250 x^2+e^{6 x} \left (2 e x-2 x^2\right )+e^{5 x} \left (-6 e x^2+6 x^3\right )+e^x \left (-150 e x^2+150 x^3\right )+e^{2 x} \left (20 x-150 x^2+30 x^4+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (4 x+30 x^2-6 x^4+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (-4 x-60 x^3+2 x^5+e \left (60 x^2-2 x^4\right )\right )\right ) \log (x)}{-125 x+e^{6 x} x-75 e^x x^2-3 e^{5 x} x^2+e^{2 x} \left (75 x-15 x^3\right )+e^{4 x} \left (-15 x+3 x^3\right )+e^{3 x} \left (30 x^2-x^4\right )} \, dx=x-\left (\frac {2}{\left (-5 e^{-x}+e^x-x\right )^2}+(-e+x)^2\right ) \log (x) \] Output:

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(32)=64\).

Time = 0.25 (sec) , antiderivative size = 129, normalized size of antiderivative = 4.03 \[ \int \frac {125 e^2-125 x-250 e x+125 x^2+e^{6 x} \left (-e^2+x+2 e x-x^2\right )+e^{5 x} \left (3 e^2 x-3 x^2-6 e x^2+3 x^3\right )+e^x \left (75 e^2 x-75 x^2-150 e x^2+75 x^3\right )+e^{2 x} \left (10+75 x-75 x^2-15 x^3+15 x^4+e^2 \left (-75+15 x^2\right )+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (-2-15 x+15 x^2+3 x^3-3 x^4+e^2 \left (15-3 x^2\right )+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (2 x+30 x^2-30 x^3-x^4+x^5+e^2 \left (-30 x+x^3\right )+e \left (60 x^2-2 x^4\right )\right )+\left (-250 e x+250 x^2+e^{6 x} \left (2 e x-2 x^2\right )+e^{5 x} \left (-6 e x^2+6 x^3\right )+e^x \left (-150 e x^2+150 x^3\right )+e^{2 x} \left (20 x-150 x^2+30 x^4+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (4 x+30 x^2-6 x^4+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (-4 x-60 x^3+2 x^5+e \left (60 x^2-2 x^4\right )\right )\right ) \log (x)}{-125 x+e^{6 x} x-75 e^x x^2-3 e^{5 x} x^2+e^{2 x} \left (75 x-15 x^3\right )+e^{4 x} \left (-15 x+3 x^3\right )+e^{3 x} \left (30 x^2-x^4\right )} \, dx=x-e^2 \log (x)-\frac {\left (-50 e x-2 e^{1+4 x} x+25 x^2+e^{4 x} x^2-20 e^{1+x} x^2+4 e^{1+3 x} x^2+10 e^x x^3-2 e^{3 x} x^3-2 e^{1+2 x} x \left (-10+x^2\right )+e^{2 x} \left (2-10 x^2+x^4\right )\right ) \log (x)}{\left (5-e^{2 x}+e^x x\right )^2} \] Input:

Integrate[(125*E^2 - 125*x - 250*E*x + 125*x^2 + E^(6*x)*(-E^2 + x + 2*E*x 
 - x^2) + E^(5*x)*(3*E^2*x - 3*x^2 - 6*E*x^2 + 3*x^3) + E^x*(75*E^2*x - 75 
*x^2 - 150*E*x^2 + 75*x^3) + E^(2*x)*(10 + 75*x - 75*x^2 - 15*x^3 + 15*x^4 
 + E^2*(-75 + 15*x^2) + E*(150*x - 30*x^3)) + E^(4*x)*(-2 - 15*x + 15*x^2 
+ 3*x^3 - 3*x^4 + E^2*(15 - 3*x^2) + E*(-30*x + 6*x^3)) + E^(3*x)*(2*x + 3 
0*x^2 - 30*x^3 - x^4 + x^5 + E^2*(-30*x + x^3) + E*(60*x^2 - 2*x^4)) + (-2 
50*E*x + 250*x^2 + E^(6*x)*(2*E*x - 2*x^2) + E^(5*x)*(-6*E*x^2 + 6*x^3) + 
E^x*(-150*E*x^2 + 150*x^3) + E^(2*x)*(20*x - 150*x^2 + 30*x^4 + E*(150*x - 
 30*x^3)) + E^(4*x)*(4*x + 30*x^2 - 6*x^4 + E*(-30*x + 6*x^3)) + E^(3*x)*( 
-4*x - 60*x^3 + 2*x^5 + E*(60*x^2 - 2*x^4)))*Log[x])/(-125*x + E^(6*x)*x - 
 75*E^x*x^2 - 3*E^(5*x)*x^2 + E^(2*x)*(75*x - 15*x^3) + E^(4*x)*(-15*x + 3 
*x^3) + E^(3*x)*(30*x^2 - x^4)),x]
 

Output:

x - E^2*Log[x] - ((-50*E*x - 2*E^(1 + 4*x)*x + 25*x^2 + E^(4*x)*x^2 - 20*E 
^(1 + x)*x^2 + 4*E^(1 + 3*x)*x^2 + 10*E^x*x^3 - 2*E^(3*x)*x^3 - 2*E^(1 + 2 
*x)*x*(-10 + x^2) + E^(2*x)*(2 - 10*x^2 + x^4))*Log[x])/(5 - E^(2*x) + E^x 
*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.

\(\displaystyle \int \frac {125 x^2+e^{6 x} \left (-x^2+2 e x+x-e^2\right )+e^{5 x} \left (3 x^3-6 e x^2-3 x^2+3 e^2 x\right )+e^x \left (75 x^3-150 e x^2-75 x^2+75 e^2 x\right )+e^{2 x} \left (15 x^4-15 x^3+e \left (150 x-30 x^3\right )-75 x^2+e^2 \left (15 x^2-75\right )+75 x+10\right )+e^{4 x} \left (-3 x^4+3 x^3+e \left (6 x^3-30 x\right )+15 x^2+e^2 \left (15-3 x^2\right )-15 x-2\right )+e^{3 x} \left (x^5-x^4-30 x^3+e^2 \left (x^3-30 x\right )+30 x^2+e \left (60 x^2-2 x^4\right )+2 x\right )+\left (250 x^2+e^{6 x} \left (2 e x-2 x^2\right )+e^{5 x} \left (6 x^3-6 e x^2\right )+e^x \left (150 x^3-150 e x^2\right )+e^{2 x} \left (30 x^4+e \left (150 x-30 x^3\right )-150 x^2+20 x\right )+e^{4 x} \left (-6 x^4+e \left (6 x^3-30 x\right )+30 x^2+4 x\right )+e^{3 x} \left (2 x^5-60 x^3+e \left (60 x^2-2 x^4\right )-4 x\right )-250 e x\right ) \log (x)-250 e x-125 x+125 e^2}{e^{2 x} \left (75 x-15 x^3\right )+e^{4 x} \left (3 x^3-15 x\right )-75 e^x x^2-3 e^{5 x} x^2+e^{3 x} \left (30 x^2-x^4\right )+e^{6 x} x-125 x} \, dx\)

\(\Big \downarrow \) 6

\(\displaystyle \int \frac {125 x^2+e^{6 x} \left (-x^2+2 e x+x-e^2\right )+e^{5 x} \left (3 x^3-6 e x^2-3 x^2+3 e^2 x\right )+e^x \left (75 x^3-150 e x^2-75 x^2+75 e^2 x\right )+e^{2 x} \left (15 x^4-15 x^3+e \left (150 x-30 x^3\right )-75 x^2+e^2 \left (15 x^2-75\right )+75 x+10\right )+e^{4 x} \left (-3 x^4+3 x^3+e \left (6 x^3-30 x\right )+15 x^2+e^2 \left (15-3 x^2\right )-15 x-2\right )+e^{3 x} \left (x^5-x^4-30 x^3+e^2 \left (x^3-30 x\right )+30 x^2+e \left (60 x^2-2 x^4\right )+2 x\right )+\left (250 x^2+e^{6 x} \left (2 e x-2 x^2\right )+e^{5 x} \left (6 x^3-6 e x^2\right )+e^x \left (150 x^3-150 e x^2\right )+e^{2 x} \left (30 x^4+e \left (150 x-30 x^3\right )-150 x^2+20 x\right )+e^{4 x} \left (-6 x^4+e \left (6 x^3-30 x\right )+30 x^2+4 x\right )+e^{3 x} \left (2 x^5-60 x^3+e \left (60 x^2-2 x^4\right )-4 x\right )-250 e x\right ) \log (x)+(-125-250 e) x+125 e^2}{e^{2 x} \left (75 x-15 x^3\right )+e^{4 x} \left (3 x^3-15 x\right )-75 e^x x^2-3 e^{5 x} x^2+e^{3 x} \left (30 x^2-x^4\right )+e^{6 x} x-125 x}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-125 x^2-e^{6 x} \left (-x^2+2 e x+x-e^2\right )-e^{5 x} \left (3 x^3-6 e x^2-3 x^2+3 e^2 x\right )-e^x \left (75 x^3-150 e x^2-75 x^2+75 e^2 x\right )-e^{2 x} \left (15 x^4-15 x^3+e \left (150 x-30 x^3\right )-75 x^2+e^2 \left (15 x^2-75\right )+75 x+10\right )-e^{4 x} \left (-3 x^4+3 x^3+e \left (6 x^3-30 x\right )+15 x^2+e^2 \left (15-3 x^2\right )-15 x-2\right )-e^{3 x} \left (x^5-x^4-30 x^3+e^2 \left (x^3-30 x\right )+30 x^2+e \left (60 x^2-2 x^4\right )+2 x\right )-\left (250 x^2+e^{6 x} \left (2 e x-2 x^2\right )+e^{5 x} \left (6 x^3-6 e x^2\right )+e^x \left (150 x^3-150 e x^2\right )+e^{2 x} \left (30 x^4+e \left (150 x-30 x^3\right )-150 x^2+20 x\right )+e^{4 x} \left (-6 x^4+e \left (6 x^3-30 x\right )+30 x^2+4 x\right )+e^{3 x} \left (2 x^5-60 x^3+e \left (60 x^2-2 x^4\right )-4 x\right )-250 e x\right ) \log (x)-(-125-250 e) x-125 e^2}{x \left (e^x x-e^{2 x}+5\right )^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {-x^2-2 x^2 \log (x)+(1+2 e) x+2 e x \log (x)-e^2}{x}+\frac {4 \left (e^x x^3-e^x x^2+5 x^2+15 e^x x-5 x-5 e^x+50\right ) \log (x)}{\left (-e^x x+e^{2 x}-5\right )^3}+\frac {2 \left (2 x^3 \log (x)+4 e^x x^2 \log (x)-2 x^2 \log (x)-e^x x-2 e^x x \log (x)+30 x \log (x)-5\right )}{x \left (e^x x-e^{2 x}+5\right )^2}-\frac {2 (2 x \log (x)-1)}{x \left (e^x x-e^{2 x}+5\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\log (x) (e-x)^2+(1+2 e) x-2 e x+200 \log (x) \int \frac {1}{\left (-e^x x+e^{2 x}-5\right )^3}dx-20 \log (x) \int \frac {e^x}{\left (-e^x x+e^{2 x}-5\right )^3}dx+60 \log (x) \int \frac {e^x x}{\left (-e^x x+e^{2 x}-5\right )^3}dx-4 \log (x) \int \frac {e^x x^2}{\left (-e^x x+e^{2 x}-5\right )^3}dx+4 \log (x) \int \frac {e^x x^3}{\left (-e^x x+e^{2 x}-5\right )^3}dx+60 \log (x) \int \frac {1}{\left (-e^x x+e^{2 x}-5\right )^2}dx-4 \log (x) \int \frac {e^x}{\left (-e^x x+e^{2 x}-5\right )^2}dx-2 \int \frac {e^x}{\left (-e^x x+e^{2 x}-5\right )^2}dx+8 \log (x) \int \frac {e^x x}{\left (-e^x x+e^{2 x}-5\right )^2}dx+4 \log (x) \int \frac {1}{-e^x x+e^{2 x}-5}dx+20 \log (x) \int \frac {x}{\left (e^x x-e^{2 x}+5\right )^3}dx-20 \log (x) \int \frac {x^2}{\left (e^x x-e^{2 x}+5\right )^3}dx-10 \int \frac {1}{x \left (e^x x-e^{2 x}+5\right )^2}dx-4 \log (x) \int \frac {x}{\left (e^x x-e^{2 x}+5\right )^2}dx+4 \log (x) \int \frac {x^2}{\left (e^x x-e^{2 x}+5\right )^2}dx+2 \int \frac {1}{x \left (e^x x-e^{2 x}+5\right )}dx-200 \int \frac {\int \frac {1}{\left (-e^x x+e^{2 x}-5\right )^3}dx}{x}dx+20 \int \frac {\int \frac {e^x}{\left (-e^x x+e^{2 x}-5\right )^3}dx}{x}dx-60 \int \frac {\int \frac {e^x x}{\left (-e^x x+e^{2 x}-5\right )^3}dx}{x}dx+4 \int \frac {\int \frac {e^x x^2}{\left (-e^x x+e^{2 x}-5\right )^3}dx}{x}dx-4 \int \frac {\int \frac {e^x x^3}{\left (-e^x x+e^{2 x}-5\right )^3}dx}{x}dx-4 \int \frac {\int \frac {1}{-e^x x+e^{2 x}-5}dx}{x}dx-20 \int \frac {\int \frac {x}{\left (e^x x-e^{2 x}+5\right )^3}dx}{x}dx+20 \int \frac {\int \frac {x^2}{\left (e^x x-e^{2 x}+5\right )^3}dx}{x}dx-60 \int \frac {\int \frac {1}{\left (e^x x-e^{2 x}+5\right )^2}dx}{x}dx+4 \int \frac {\int \frac {e^x}{\left (e^x x-e^{2 x}+5\right )^2}dx}{x}dx+4 \int \frac {\int \frac {x}{\left (e^x x-e^{2 x}+5\right )^2}dx}{x}dx-8 \int \frac {\int \frac {e^x x}{\left (e^x x-e^{2 x}+5\right )^2}dx}{x}dx-4 \int \frac {\int \frac {x^2}{\left (e^x x-e^{2 x}+5\right )^2}dx}{x}dx\)

Input:

Int[(125*E^2 - 125*x - 250*E*x + 125*x^2 + E^(6*x)*(-E^2 + x + 2*E*x - x^2 
) + E^(5*x)*(3*E^2*x - 3*x^2 - 6*E*x^2 + 3*x^3) + E^x*(75*E^2*x - 75*x^2 - 
 150*E*x^2 + 75*x^3) + E^(2*x)*(10 + 75*x - 75*x^2 - 15*x^3 + 15*x^4 + E^2 
*(-75 + 15*x^2) + E*(150*x - 30*x^3)) + E^(4*x)*(-2 - 15*x + 15*x^2 + 3*x^ 
3 - 3*x^4 + E^2*(15 - 3*x^2) + E*(-30*x + 6*x^3)) + E^(3*x)*(2*x + 30*x^2 
- 30*x^3 - x^4 + x^5 + E^2*(-30*x + x^3) + E*(60*x^2 - 2*x^4)) + (-250*E*x 
 + 250*x^2 + E^(6*x)*(2*E*x - 2*x^2) + E^(5*x)*(-6*E*x^2 + 6*x^3) + E^x*(- 
150*E*x^2 + 150*x^3) + E^(2*x)*(20*x - 150*x^2 + 30*x^4 + E*(150*x - 30*x^ 
3)) + E^(4*x)*(4*x + 30*x^2 - 6*x^4 + E*(-30*x + 6*x^3)) + E^(3*x)*(-4*x - 
 60*x^3 + 2*x^5 + E*(60*x^2 - 2*x^4)))*Log[x])/(-125*x + E^(6*x)*x - 75*E^ 
x*x^2 - 3*E^(5*x)*x^2 + E^(2*x)*(75*x - 15*x^3) + E^(4*x)*(-15*x + 3*x^3) 
+ E^(3*x)*(30*x^2 - x^4)),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(134\) vs. \(2(31)=62\).

Time = 0.06 (sec) , antiderivative size = 135, normalized size of antiderivative = 4.22

\[\frac {\left (2 x^{3} {\mathrm e}^{1+2 x}-4 x^{2} {\mathrm e}^{1+3 x}+2 x \,{\mathrm e}^{1+4 x}-{\mathrm e}^{2 x} x^{4}+2 x^{3} {\mathrm e}^{3 x}-x^{2} {\mathrm e}^{4 x}+20 x^{2} {\mathrm e}^{1+x}-20 x \,{\mathrm e}^{1+2 x}-10 \,{\mathrm e}^{x} x^{3}+10 \,{\mathrm e}^{2 x} x^{2}+50 x \,{\mathrm e}-25 x^{2}-2 \,{\mathrm e}^{2 x}\right ) \ln \left (x \right )}{\left ({\mathrm e}^{x} x -{\mathrm e}^{2 x}+5\right )^{2}}+x -{\mathrm e}^{2} \ln \left (x \right )\]

Input:

int((((2*x*exp(1)-2*x^2)*exp(x)^6+(-6*x^2*exp(1)+6*x^3)*exp(x)^5+((6*x^3-3 
0*x)*exp(1)-6*x^4+30*x^2+4*x)*exp(x)^4+((-2*x^4+60*x^2)*exp(1)+2*x^5-60*x^ 
3-4*x)*exp(x)^3+((-30*x^3+150*x)*exp(1)+30*x^4-150*x^2+20*x)*exp(x)^2+(-15 
0*x^2*exp(1)+150*x^3)*exp(x)-250*x*exp(1)+250*x^2)*ln(x)+(-exp(1)^2+2*x*ex 
p(1)-x^2+x)*exp(x)^6+(3*x*exp(1)^2-6*x^2*exp(1)+3*x^3-3*x^2)*exp(x)^5+((-3 
*x^2+15)*exp(1)^2+(6*x^3-30*x)*exp(1)-3*x^4+3*x^3+15*x^2-15*x-2)*exp(x)^4+ 
((x^3-30*x)*exp(1)^2+(-2*x^4+60*x^2)*exp(1)+x^5-x^4-30*x^3+30*x^2+2*x)*exp 
(x)^3+((15*x^2-75)*exp(1)^2+(-30*x^3+150*x)*exp(1)+15*x^4-15*x^3-75*x^2+75 
*x+10)*exp(x)^2+(75*x*exp(1)^2-150*x^2*exp(1)+75*x^3-75*x^2)*exp(x)+125*ex 
p(1)^2-250*x*exp(1)+125*x^2-125*x)/(x*exp(x)^6-3*x^2*exp(x)^5+(3*x^3-15*x) 
*exp(x)^4+(-x^4+30*x^2)*exp(x)^3+(-15*x^3+75*x)*exp(x)^2-75*exp(x)*x^2-125 
*x),x)
 

Output:

(2*x^3*exp(1+2*x)-4*x^2*exp(1+3*x)+2*x*exp(1+4*x)-exp(2*x)*x^4+2*x^3*exp(3 
*x)-x^2*exp(4*x)+20*x^2*exp(1+x)-20*x*exp(1+2*x)-10*exp(x)*x^3+10*exp(2*x) 
*x^2+50*x*exp(1)-25*x^2-2*exp(2*x))/(exp(x)*x-exp(2*x)+5)^2*ln(x)+x-exp(2) 
*ln(x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (31) = 62\).

Time = 0.11 (sec) , antiderivative size = 182, normalized size of antiderivative = 5.69 \[ \int \frac {125 e^2-125 x-250 e x+125 x^2+e^{6 x} \left (-e^2+x+2 e x-x^2\right )+e^{5 x} \left (3 e^2 x-3 x^2-6 e x^2+3 x^3\right )+e^x \left (75 e^2 x-75 x^2-150 e x^2+75 x^3\right )+e^{2 x} \left (10+75 x-75 x^2-15 x^3+15 x^4+e^2 \left (-75+15 x^2\right )+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (-2-15 x+15 x^2+3 x^3-3 x^4+e^2 \left (15-3 x^2\right )+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (2 x+30 x^2-30 x^3-x^4+x^5+e^2 \left (-30 x+x^3\right )+e \left (60 x^2-2 x^4\right )\right )+\left (-250 e x+250 x^2+e^{6 x} \left (2 e x-2 x^2\right )+e^{5 x} \left (-6 e x^2+6 x^3\right )+e^x \left (-150 e x^2+150 x^3\right )+e^{2 x} \left (20 x-150 x^2+30 x^4+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (4 x+30 x^2-6 x^4+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (-4 x-60 x^3+2 x^5+e \left (60 x^2-2 x^4\right )\right )\right ) \log (x)}{-125 x+e^{6 x} x-75 e^x x^2-3 e^{5 x} x^2+e^{2 x} \left (75 x-15 x^3\right )+e^{4 x} \left (-15 x+3 x^3\right )+e^{3 x} \left (30 x^2-x^4\right )} \, dx=\frac {2 \, x^{2} e^{\left (3 \, x\right )} - 10 \, x^{2} e^{x} - x e^{\left (4 \, x\right )} - {\left (x^{3} - 10 \, x\right )} e^{\left (2 \, x\right )} + {\left (25 \, x^{2} - 50 \, x e + {\left (x^{2} - 2 \, x e + e^{2}\right )} e^{\left (4 \, x\right )} - 2 \, {\left (x^{3} - 2 \, x^{2} e + x e^{2}\right )} e^{\left (3 \, x\right )} + {\left (x^{4} - 10 \, x^{2} + {\left (x^{2} - 10\right )} e^{2} - 2 \, {\left (x^{3} - 10 \, x\right )} e + 2\right )} e^{\left (2 \, x\right )} + 10 \, {\left (x^{3} - 2 \, x^{2} e + x e^{2}\right )} e^{x} + 25 \, e^{2}\right )} \log \left (x\right ) - 25 \, x}{2 \, x e^{\left (3 \, x\right )} - {\left (x^{2} - 10\right )} e^{\left (2 \, x\right )} - 10 \, x e^{x} - e^{\left (4 \, x\right )} - 25} \] Input:

integrate((((2*exp(1)*x-2*x^2)*exp(x)^6+(-6*x^2*exp(1)+6*x^3)*exp(x)^5+((6 
*x^3-30*x)*exp(1)-6*x^4+30*x^2+4*x)*exp(x)^4+((-2*x^4+60*x^2)*exp(1)+2*x^5 
-60*x^3-4*x)*exp(x)^3+((-30*x^3+150*x)*exp(1)+30*x^4-150*x^2+20*x)*exp(x)^ 
2+(-150*x^2*exp(1)+150*x^3)*exp(x)-250*exp(1)*x+250*x^2)*log(x)+(-exp(1)^2 
+2*exp(1)*x-x^2+x)*exp(x)^6+(3*x*exp(1)^2-6*x^2*exp(1)+3*x^3-3*x^2)*exp(x) 
^5+((-3*x^2+15)*exp(1)^2+(6*x^3-30*x)*exp(1)-3*x^4+3*x^3+15*x^2-15*x-2)*ex 
p(x)^4+((x^3-30*x)*exp(1)^2+(-2*x^4+60*x^2)*exp(1)+x^5-x^4-30*x^3+30*x^2+2 
*x)*exp(x)^3+((15*x^2-75)*exp(1)^2+(-30*x^3+150*x)*exp(1)+15*x^4-15*x^3-75 
*x^2+75*x+10)*exp(x)^2+(75*x*exp(1)^2-150*x^2*exp(1)+75*x^3-75*x^2)*exp(x) 
+125*exp(1)^2-250*exp(1)*x+125*x^2-125*x)/(x*exp(x)^6-3*x^2*exp(x)^5+(3*x^ 
3-15*x)*exp(x)^4+(-x^4+30*x^2)*exp(x)^3+(-15*x^3+75*x)*exp(x)^2-75*exp(x)* 
x^2-125*x),x, algorithm="fricas")
 

Output:

(2*x^2*e^(3*x) - 10*x^2*e^x - x*e^(4*x) - (x^3 - 10*x)*e^(2*x) + (25*x^2 - 
 50*x*e + (x^2 - 2*x*e + e^2)*e^(4*x) - 2*(x^3 - 2*x^2*e + x*e^2)*e^(3*x) 
+ (x^4 - 10*x^2 + (x^2 - 10)*e^2 - 2*(x^3 - 10*x)*e + 2)*e^(2*x) + 10*(x^3 
 - 2*x^2*e + x*e^2)*e^x + 25*e^2)*log(x) - 25*x)/(2*x*e^(3*x) - (x^2 - 10) 
*e^(2*x) - 10*x*e^x - e^(4*x) - 25)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (24) = 48\).

Time = 0.32 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.97 \[ \int \frac {125 e^2-125 x-250 e x+125 x^2+e^{6 x} \left (-e^2+x+2 e x-x^2\right )+e^{5 x} \left (3 e^2 x-3 x^2-6 e x^2+3 x^3\right )+e^x \left (75 e^2 x-75 x^2-150 e x^2+75 x^3\right )+e^{2 x} \left (10+75 x-75 x^2-15 x^3+15 x^4+e^2 \left (-75+15 x^2\right )+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (-2-15 x+15 x^2+3 x^3-3 x^4+e^2 \left (15-3 x^2\right )+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (2 x+30 x^2-30 x^3-x^4+x^5+e^2 \left (-30 x+x^3\right )+e \left (60 x^2-2 x^4\right )\right )+\left (-250 e x+250 x^2+e^{6 x} \left (2 e x-2 x^2\right )+e^{5 x} \left (-6 e x^2+6 x^3\right )+e^x \left (-150 e x^2+150 x^3\right )+e^{2 x} \left (20 x-150 x^2+30 x^4+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (4 x+30 x^2-6 x^4+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (-4 x-60 x^3+2 x^5+e \left (60 x^2-2 x^4\right )\right )\right ) \log (x)}{-125 x+e^{6 x} x-75 e^x x^2-3 e^{5 x} x^2+e^{2 x} \left (75 x-15 x^3\right )+e^{4 x} \left (-15 x+3 x^3\right )+e^{3 x} \left (30 x^2-x^4\right )} \, dx=x + \left (- x^{2} + 2 e x\right ) \log {\left (x \right )} - e^{2} \log {\left (x \right )} - \frac {2 e^{2 x} \log {\left (x \right )}}{- 2 x e^{3 x} + 10 x e^{x} + \left (x^{2} - 10\right ) e^{2 x} + e^{4 x} + 25} \] Input:

integrate((((2*exp(1)*x-2*x**2)*exp(x)**6+(-6*x**2*exp(1)+6*x**3)*exp(x)** 
5+((6*x**3-30*x)*exp(1)-6*x**4+30*x**2+4*x)*exp(x)**4+((-2*x**4+60*x**2)*e 
xp(1)+2*x**5-60*x**3-4*x)*exp(x)**3+((-30*x**3+150*x)*exp(1)+30*x**4-150*x 
**2+20*x)*exp(x)**2+(-150*x**2*exp(1)+150*x**3)*exp(x)-250*exp(1)*x+250*x* 
*2)*ln(x)+(-exp(1)**2+2*exp(1)*x-x**2+x)*exp(x)**6+(3*x*exp(1)**2-6*x**2*e 
xp(1)+3*x**3-3*x**2)*exp(x)**5+((-3*x**2+15)*exp(1)**2+(6*x**3-30*x)*exp(1 
)-3*x**4+3*x**3+15*x**2-15*x-2)*exp(x)**4+((x**3-30*x)*exp(1)**2+(-2*x**4+ 
60*x**2)*exp(1)+x**5-x**4-30*x**3+30*x**2+2*x)*exp(x)**3+((15*x**2-75)*exp 
(1)**2+(-30*x**3+150*x)*exp(1)+15*x**4-15*x**3-75*x**2+75*x+10)*exp(x)**2+ 
(75*x*exp(1)**2-150*x**2*exp(1)+75*x**3-75*x**2)*exp(x)+125*exp(1)**2-250* 
exp(1)*x+125*x**2-125*x)/(x*exp(x)**6-3*x**2*exp(x)**5+(3*x**3-15*x)*exp(x 
)**4+(-x**4+30*x**2)*exp(x)**3+(-15*x**3+75*x)*exp(x)**2-75*exp(x)*x**2-12 
5*x),x)
 

Output:

x + (-x**2 + 2*E*x)*log(x) - exp(2)*log(x) - 2*exp(2*x)*log(x)/(-2*x*exp(3 
*x) + 10*x*exp(x) + (x**2 - 10)*exp(2*x) + exp(4*x) + 25)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 178 vs. \(2 (31) = 62\).

Time = 0.19 (sec) , antiderivative size = 178, normalized size of antiderivative = 5.56 \[ \int \frac {125 e^2-125 x-250 e x+125 x^2+e^{6 x} \left (-e^2+x+2 e x-x^2\right )+e^{5 x} \left (3 e^2 x-3 x^2-6 e x^2+3 x^3\right )+e^x \left (75 e^2 x-75 x^2-150 e x^2+75 x^3\right )+e^{2 x} \left (10+75 x-75 x^2-15 x^3+15 x^4+e^2 \left (-75+15 x^2\right )+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (-2-15 x+15 x^2+3 x^3-3 x^4+e^2 \left (15-3 x^2\right )+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (2 x+30 x^2-30 x^3-x^4+x^5+e^2 \left (-30 x+x^3\right )+e \left (60 x^2-2 x^4\right )\right )+\left (-250 e x+250 x^2+e^{6 x} \left (2 e x-2 x^2\right )+e^{5 x} \left (-6 e x^2+6 x^3\right )+e^x \left (-150 e x^2+150 x^3\right )+e^{2 x} \left (20 x-150 x^2+30 x^4+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (4 x+30 x^2-6 x^4+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (-4 x-60 x^3+2 x^5+e \left (60 x^2-2 x^4\right )\right )\right ) \log (x)}{-125 x+e^{6 x} x-75 e^x x^2-3 e^{5 x} x^2+e^{2 x} \left (75 x-15 x^3\right )+e^{4 x} \left (-15 x+3 x^3\right )+e^{3 x} \left (30 x^2-x^4\right )} \, dx=\frac {{\left ({\left (x^{2} - 2 \, x e + e^{2}\right )} \log \left (x\right ) - x\right )} e^{\left (4 \, x\right )} + 2 \, {\left (x^{2} - {\left (x^{3} - 2 \, x^{2} e + x e^{2}\right )} \log \left (x\right )\right )} e^{\left (3 \, x\right )} - {\left (x^{3} - {\left (x^{4} - 2 \, x^{3} e + x^{2} {\left (e^{2} - 10\right )} + 20 \, x e - 10 \, e^{2} + 2\right )} \log \left (x\right ) - 10 \, x\right )} e^{\left (2 \, x\right )} - 10 \, {\left (x^{2} - {\left (x^{3} - 2 \, x^{2} e + x e^{2}\right )} \log \left (x\right )\right )} e^{x} + 25 \, {\left (x^{2} - 2 \, x e + e^{2}\right )} \log \left (x\right ) - 25 \, x}{2 \, x e^{\left (3 \, x\right )} - {\left (x^{2} - 10\right )} e^{\left (2 \, x\right )} - 10 \, x e^{x} - e^{\left (4 \, x\right )} - 25} \] Input:

integrate((((2*exp(1)*x-2*x^2)*exp(x)^6+(-6*x^2*exp(1)+6*x^3)*exp(x)^5+((6 
*x^3-30*x)*exp(1)-6*x^4+30*x^2+4*x)*exp(x)^4+((-2*x^4+60*x^2)*exp(1)+2*x^5 
-60*x^3-4*x)*exp(x)^3+((-30*x^3+150*x)*exp(1)+30*x^4-150*x^2+20*x)*exp(x)^ 
2+(-150*x^2*exp(1)+150*x^3)*exp(x)-250*exp(1)*x+250*x^2)*log(x)+(-exp(1)^2 
+2*exp(1)*x-x^2+x)*exp(x)^6+(3*x*exp(1)^2-6*x^2*exp(1)+3*x^3-3*x^2)*exp(x) 
^5+((-3*x^2+15)*exp(1)^2+(6*x^3-30*x)*exp(1)-3*x^4+3*x^3+15*x^2-15*x-2)*ex 
p(x)^4+((x^3-30*x)*exp(1)^2+(-2*x^4+60*x^2)*exp(1)+x^5-x^4-30*x^3+30*x^2+2 
*x)*exp(x)^3+((15*x^2-75)*exp(1)^2+(-30*x^3+150*x)*exp(1)+15*x^4-15*x^3-75 
*x^2+75*x+10)*exp(x)^2+(75*x*exp(1)^2-150*x^2*exp(1)+75*x^3-75*x^2)*exp(x) 
+125*exp(1)^2-250*exp(1)*x+125*x^2-125*x)/(x*exp(x)^6-3*x^2*exp(x)^5+(3*x^ 
3-15*x)*exp(x)^4+(-x^4+30*x^2)*exp(x)^3+(-15*x^3+75*x)*exp(x)^2-75*exp(x)* 
x^2-125*x),x, algorithm="maxima")
 

Output:

(((x^2 - 2*x*e + e^2)*log(x) - x)*e^(4*x) + 2*(x^2 - (x^3 - 2*x^2*e + x*e^ 
2)*log(x))*e^(3*x) - (x^3 - (x^4 - 2*x^3*e + x^2*(e^2 - 10) + 20*x*e - 10* 
e^2 + 2)*log(x) - 10*x)*e^(2*x) - 10*(x^2 - (x^3 - 2*x^2*e + x*e^2)*log(x) 
)*e^x + 25*(x^2 - 2*x*e + e^2)*log(x) - 25*x)/(2*x*e^(3*x) - (x^2 - 10)*e^ 
(2*x) - 10*x*e^x - e^(4*x) - 25)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1806 vs. \(2 (31) = 62\).

Time = 0.32 (sec) , antiderivative size = 1806, normalized size of antiderivative = 56.44 \[ \int \frac {125 e^2-125 x-250 e x+125 x^2+e^{6 x} \left (-e^2+x+2 e x-x^2\right )+e^{5 x} \left (3 e^2 x-3 x^2-6 e x^2+3 x^3\right )+e^x \left (75 e^2 x-75 x^2-150 e x^2+75 x^3\right )+e^{2 x} \left (10+75 x-75 x^2-15 x^3+15 x^4+e^2 \left (-75+15 x^2\right )+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (-2-15 x+15 x^2+3 x^3-3 x^4+e^2 \left (15-3 x^2\right )+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (2 x+30 x^2-30 x^3-x^4+x^5+e^2 \left (-30 x+x^3\right )+e \left (60 x^2-2 x^4\right )\right )+\left (-250 e x+250 x^2+e^{6 x} \left (2 e x-2 x^2\right )+e^{5 x} \left (-6 e x^2+6 x^3\right )+e^x \left (-150 e x^2+150 x^3\right )+e^{2 x} \left (20 x-150 x^2+30 x^4+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (4 x+30 x^2-6 x^4+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (-4 x-60 x^3+2 x^5+e \left (60 x^2-2 x^4\right )\right )\right ) \log (x)}{-125 x+e^{6 x} x-75 e^x x^2-3 e^{5 x} x^2+e^{2 x} \left (75 x-15 x^3\right )+e^{4 x} \left (-15 x+3 x^3\right )+e^{3 x} \left (30 x^2-x^4\right )} \, dx=\text {Too large to display} \] Input:

integrate((((2*exp(1)*x-2*x^2)*exp(x)^6+(-6*x^2*exp(1)+6*x^3)*exp(x)^5+((6 
*x^3-30*x)*exp(1)-6*x^4+30*x^2+4*x)*exp(x)^4+((-2*x^4+60*x^2)*exp(1)+2*x^5 
-60*x^3-4*x)*exp(x)^3+((-30*x^3+150*x)*exp(1)+30*x^4-150*x^2+20*x)*exp(x)^ 
2+(-150*x^2*exp(1)+150*x^3)*exp(x)-250*exp(1)*x+250*x^2)*log(x)+(-exp(1)^2 
+2*exp(1)*x-x^2+x)*exp(x)^6+(3*x*exp(1)^2-6*x^2*exp(1)+3*x^3-3*x^2)*exp(x) 
^5+((-3*x^2+15)*exp(1)^2+(6*x^3-30*x)*exp(1)-3*x^4+3*x^3+15*x^2-15*x-2)*ex 
p(x)^4+((x^3-30*x)*exp(1)^2+(-2*x^4+60*x^2)*exp(1)+x^5-x^4-30*x^3+30*x^2+2 
*x)*exp(x)^3+((15*x^2-75)*exp(1)^2+(-30*x^3+150*x)*exp(1)+15*x^4-15*x^3-75 
*x^2+75*x+10)*exp(x)^2+(75*x*exp(1)^2-150*x^2*exp(1)+75*x^3-75*x^2)*exp(x) 
+125*exp(1)^2-250*exp(1)*x+125*x^2-125*x)/(x*exp(x)^6-3*x^2*exp(x)^5+(3*x^ 
3-15*x)*exp(x)^4+(-x^4+30*x^2)*exp(x)^3+(-15*x^3+75*x)*exp(x)^2-75*exp(x)* 
x^2-125*x),x, algorithm="giac")
 

Output:

-1/2*(2*x^11*e^(2*x)*log(x) - 4*x^10*e^(3*x)*log(x) - 4*x^10*e^(2*x + 1)*l 
og(x) + 20*x^10*e^x*log(x) - 2*x^10*e^(2*x) + 2*x^9*e^(4*x)*log(x) + 94*x^ 
9*e^(2*x)*log(x) + 8*x^9*e^(3*x + 1)*log(x) + 2*x^9*e^(2*x + 2)*log(x) - 4 
0*x^9*e^(x + 1)*log(x) + 20*x^9*e^x*log(x) + 4*x^9*e^(3*x) - 10*x^9*e^x + 
50*x^9*log(x) - 100*x^8*e*log(x) - 228*x^8*e^(3*x)*log(x) - 40*x^8*e^(2*x) 
*log(x) - 4*x^8*e^(4*x + 1)*log(x) - 4*x^8*e^(3*x + 2)*log(x) - 188*x^8*e^ 
(2*x + 1)*log(x) + 20*x^8*e^(x + 2)*log(x) - 20*x^8*e^(x + 1)*log(x) + 111 
0*x^8*e^x*log(x) - 2*x^8*e^(4*x) - 114*x^8*e^(2*x) - 20*x^8*e^(x + 1) - 35 
*x^8*e^x + 150*x^8*log(x) + 50*x^7*e^2*log(x) - 150*x^7*e*log(x) + 114*x^7 
*e^(4*x)*log(x) + 20*x^7*e^(3*x)*log(x) + 1090*x^7*e^(2*x)*log(x) + 2*x^7* 
e^(4*x + 2)*log(x) + 456*x^7*e^(3*x + 1)*log(x) + 94*x^7*e^(2*x + 2)*log(x 
) + 40*x^7*e^(2*x + 1)*log(x) - 2260*x^7*e^(x + 1)*log(x) + 970*x^7*e^x*lo 
g(x) + 25*x^8 - 150*x^7*e + 238*x^7*e^(3*x) + 70*x^7*e^(2*x) + 40*x^7*e^(2 
*x + 1) + 30*x^7*e^(x + 1) - 625*x^7*e^x + 2650*x^7*log(x) - 5550*x^6*e*lo 
g(x) - 4362*x^6*e^(3*x)*log(x) - 2190*x^6*e^(2*x)*log(x) - 228*x^6*e^(4*x 
+ 1)*log(x) - 228*x^6*e^(3*x + 2)*log(x) - 20*x^6*e^(3*x + 1)*log(x) - 209 
2*x^6*e^(2*x + 1)*log(x) + 1140*x^6*e^(x + 2)*log(x) - 950*x^6*e^(x + 1)*l 
og(x) + 20230*x^6*e^x*log(x) - 225*x^7 + 200*x^6*e - 114*x^6*e^(4*x) - 35* 
x^6*e^(3*x) - 2181*x^6*e^(2*x) - 20*x^6*e^(3*x + 1) - 60*x^6*e^(2*x + 1) - 
 970*x^6*e^(x + 1) - 1590*x^6*e^x + 8000*x^6*log(x) + 2850*x^5*e^2*log(...
 

Mupad [B] (verification not implemented)

Time = 3.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 3.94 \[ \int \frac {125 e^2-125 x-250 e x+125 x^2+e^{6 x} \left (-e^2+x+2 e x-x^2\right )+e^{5 x} \left (3 e^2 x-3 x^2-6 e x^2+3 x^3\right )+e^x \left (75 e^2 x-75 x^2-150 e x^2+75 x^3\right )+e^{2 x} \left (10+75 x-75 x^2-15 x^3+15 x^4+e^2 \left (-75+15 x^2\right )+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (-2-15 x+15 x^2+3 x^3-3 x^4+e^2 \left (15-3 x^2\right )+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (2 x+30 x^2-30 x^3-x^4+x^5+e^2 \left (-30 x+x^3\right )+e \left (60 x^2-2 x^4\right )\right )+\left (-250 e x+250 x^2+e^{6 x} \left (2 e x-2 x^2\right )+e^{5 x} \left (-6 e x^2+6 x^3\right )+e^x \left (-150 e x^2+150 x^3\right )+e^{2 x} \left (20 x-150 x^2+30 x^4+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (4 x+30 x^2-6 x^4+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (-4 x-60 x^3+2 x^5+e \left (60 x^2-2 x^4\right )\right )\right ) \log (x)}{-125 x+e^{6 x} x-75 e^x x^2-3 e^{5 x} x^2+e^{2 x} \left (75 x-15 x^3\right )+e^{4 x} \left (-15 x+3 x^3\right )+e^{3 x} \left (30 x^2-x^4\right )} \, dx=x-{\mathrm {e}}^2\,\ln \left (x\right )+\frac {\ln \left (x\right )\,\left (50\,x\,\mathrm {e}+{\mathrm {e}}^{2\,x}\,\left (\left (x^2-10\right )\,\left (2\,x\,\mathrm {e}-x^2\right )-2\right )+{\mathrm {e}}^{4\,x}\,\left (2\,x\,\mathrm {e}-x^2\right )-25\,x^2+10\,x\,{\mathrm {e}}^x\,\left (2\,x\,\mathrm {e}-x^2\right )-2\,x\,{\mathrm {e}}^{3\,x}\,\left (2\,x\,\mathrm {e}-x^2\right )\right )}{{\mathrm {e}}^{4\,x}-2\,x\,{\mathrm {e}}^{3\,x}+10\,x\,{\mathrm {e}}^x+{\mathrm {e}}^{2\,x}\,\left (x^2-10\right )+25} \] Input:

int(-(125*exp(2) - 125*x - exp(4*x)*(15*x + exp(1)*(30*x - 6*x^3) + exp(2) 
*(3*x^2 - 15) - 15*x^2 - 3*x^3 + 3*x^4 + 2) + exp(2*x)*(75*x + exp(1)*(150 
*x - 30*x^3) + exp(2)*(15*x^2 - 75) - 75*x^2 - 15*x^3 + 15*x^4 + 10) + exp 
(3*x)*(2*x - exp(2)*(30*x - x^3) + exp(1)*(60*x^2 - 2*x^4) + 30*x^2 - 30*x 
^3 - x^4 + x^5) - 250*x*exp(1) + exp(5*x)*(3*x*exp(2) - 6*x^2*exp(1) - 3*x 
^2 + 3*x^3) - log(x)*(250*x*exp(1) - exp(6*x)*(2*x*exp(1) - 2*x^2) + exp(x 
)*(150*x^2*exp(1) - 150*x^3) - exp(4*x)*(4*x - exp(1)*(30*x - 6*x^3) + 30* 
x^2 - 6*x^4) - exp(2*x)*(20*x + exp(1)*(150*x - 30*x^3) - 150*x^2 + 30*x^4 
) + exp(5*x)*(6*x^2*exp(1) - 6*x^3) + exp(3*x)*(4*x - exp(1)*(60*x^2 - 2*x 
^4) + 60*x^3 - 2*x^5) - 250*x^2) + exp(6*x)*(x - exp(2) + 2*x*exp(1) - x^2 
) + 125*x^2 + exp(x)*(75*x*exp(2) - 150*x^2*exp(1) - 75*x^2 + 75*x^3))/(12 
5*x + exp(4*x)*(15*x - 3*x^3) - exp(2*x)*(75*x - 15*x^3) - x*exp(6*x) + 75 
*x^2*exp(x) - exp(3*x)*(30*x^2 - x^4) + 3*x^2*exp(5*x)),x)
 

Output:

x - exp(2)*log(x) + (log(x)*(50*x*exp(1) + exp(2*x)*((x^2 - 10)*(2*x*exp(1 
) - x^2) - 2) + exp(4*x)*(2*x*exp(1) - x^2) - 25*x^2 + 10*x*exp(x)*(2*x*ex 
p(1) - x^2) - 2*x*exp(3*x)*(2*x*exp(1) - x^2)))/(exp(4*x) - 2*x*exp(3*x) + 
 10*x*exp(x) + exp(2*x)*(x^2 - 10) + 25)
 

Reduce [B] (verification not implemented)

Time = 0.57 (sec) , antiderivative size = 295, normalized size of antiderivative = 9.22 \[ \int \frac {125 e^2-125 x-250 e x+125 x^2+e^{6 x} \left (-e^2+x+2 e x-x^2\right )+e^{5 x} \left (3 e^2 x-3 x^2-6 e x^2+3 x^3\right )+e^x \left (75 e^2 x-75 x^2-150 e x^2+75 x^3\right )+e^{2 x} \left (10+75 x-75 x^2-15 x^3+15 x^4+e^2 \left (-75+15 x^2\right )+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (-2-15 x+15 x^2+3 x^3-3 x^4+e^2 \left (15-3 x^2\right )+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (2 x+30 x^2-30 x^3-x^4+x^5+e^2 \left (-30 x+x^3\right )+e \left (60 x^2-2 x^4\right )\right )+\left (-250 e x+250 x^2+e^{6 x} \left (2 e x-2 x^2\right )+e^{5 x} \left (-6 e x^2+6 x^3\right )+e^x \left (-150 e x^2+150 x^3\right )+e^{2 x} \left (20 x-150 x^2+30 x^4+e \left (150 x-30 x^3\right )\right )+e^{4 x} \left (4 x+30 x^2-6 x^4+e \left (-30 x+6 x^3\right )\right )+e^{3 x} \left (-4 x-60 x^3+2 x^5+e \left (60 x^2-2 x^4\right )\right )\right ) \log (x)}{-125 x+e^{6 x} x-75 e^x x^2-3 e^{5 x} x^2+e^{2 x} \left (75 x-15 x^3\right )+e^{4 x} \left (-15 x+3 x^3\right )+e^{3 x} \left (30 x^2-x^4\right )} \, dx=\frac {25 x -e^{4 x} \mathrm {log}\left (x \right ) x^{2}+50 \,\mathrm {log}\left (x \right ) e x +e^{2 x} x^{3}+2 e^{3 x} \mathrm {log}\left (x \right ) x^{3}+10 e^{2 x} \mathrm {log}\left (x \right ) e^{2}-e^{2 x} \mathrm {log}\left (x \right ) x^{4}+2 e^{4 x} \mathrm {log}\left (x \right ) e x +2 e^{3 x} \mathrm {log}\left (x \right ) e^{2} x -4 e^{3 x} \mathrm {log}\left (x \right ) e \,x^{2}+2 e^{2 x} \mathrm {log}\left (x \right ) e \,x^{3}-20 e^{2 x} \mathrm {log}\left (x \right ) e x -10 e^{x} \mathrm {log}\left (x \right ) e^{2} x +20 e^{x} \mathrm {log}\left (x \right ) e \,x^{2}+e^{4 x} x +10 e^{2 x} \mathrm {log}\left (x \right ) x^{2}+10 e^{x} x^{2}-10 e^{x} \mathrm {log}\left (x \right ) x^{3}-25 \,\mathrm {log}\left (x \right ) x^{2}-10 e^{2 x} x -25 \,\mathrm {log}\left (x \right ) e^{2}-e^{4 x} \mathrm {log}\left (x \right ) e^{2}-e^{2 x} \mathrm {log}\left (x \right ) e^{2} x^{2}-2 e^{3 x} x^{2}-2 e^{2 x} \mathrm {log}\left (x \right )}{e^{4 x}-2 e^{3 x} x +e^{2 x} x^{2}-10 e^{2 x}+10 e^{x} x +25} \] Input:

int((((2*exp(1)*x-2*x^2)*exp(x)^6+(-6*x^2*exp(1)+6*x^3)*exp(x)^5+((6*x^3-3 
0*x)*exp(1)-6*x^4+30*x^2+4*x)*exp(x)^4+((-2*x^4+60*x^2)*exp(1)+2*x^5-60*x^ 
3-4*x)*exp(x)^3+((-30*x^3+150*x)*exp(1)+30*x^4-150*x^2+20*x)*exp(x)^2+(-15 
0*x^2*exp(1)+150*x^3)*exp(x)-250*exp(1)*x+250*x^2)*log(x)+(-exp(1)^2+2*exp 
(1)*x-x^2+x)*exp(x)^6+(3*x*exp(1)^2-6*x^2*exp(1)+3*x^3-3*x^2)*exp(x)^5+((- 
3*x^2+15)*exp(1)^2+(6*x^3-30*x)*exp(1)-3*x^4+3*x^3+15*x^2-15*x-2)*exp(x)^4 
+((x^3-30*x)*exp(1)^2+(-2*x^4+60*x^2)*exp(1)+x^5-x^4-30*x^3+30*x^2+2*x)*ex 
p(x)^3+((15*x^2-75)*exp(1)^2+(-30*x^3+150*x)*exp(1)+15*x^4-15*x^3-75*x^2+7 
5*x+10)*exp(x)^2+(75*x*exp(1)^2-150*x^2*exp(1)+75*x^3-75*x^2)*exp(x)+125*e 
xp(1)^2-250*exp(1)*x+125*x^2-125*x)/(x*exp(x)^6-3*x^2*exp(x)^5+(3*x^3-15*x 
)*exp(x)^4+(-x^4+30*x^2)*exp(x)^3+(-15*x^3+75*x)*exp(x)^2-75*exp(x)*x^2-12 
5*x),x)
 

Output:

( - e**(4*x)*log(x)*e**2 + 2*e**(4*x)*log(x)*e*x - e**(4*x)*log(x)*x**2 + 
e**(4*x)*x + 2*e**(3*x)*log(x)*e**2*x - 4*e**(3*x)*log(x)*e*x**2 + 2*e**(3 
*x)*log(x)*x**3 - 2*e**(3*x)*x**2 - e**(2*x)*log(x)*e**2*x**2 + 10*e**(2*x 
)*log(x)*e**2 + 2*e**(2*x)*log(x)*e*x**3 - 20*e**(2*x)*log(x)*e*x - e**(2* 
x)*log(x)*x**4 + 10*e**(2*x)*log(x)*x**2 - 2*e**(2*x)*log(x) + e**(2*x)*x* 
*3 - 10*e**(2*x)*x - 10*e**x*log(x)*e**2*x + 20*e**x*log(x)*e*x**2 - 10*e* 
*x*log(x)*x**3 + 10*e**x*x**2 - 25*log(x)*e**2 + 50*log(x)*e*x - 25*log(x) 
*x**2 + 25*x)/(e**(4*x) - 2*e**(3*x)*x + e**(2*x)*x**2 - 10*e**(2*x) + 10* 
e**x*x + 25)