\(\int \frac {-75-150 e^{2 x}-150 x^2+e^{2 x/3} (-15+10 x)+e^x (75+5 e^{2 x/3}+300 x)}{100+4 e^{4 x/3}-400 x^2+400 e^{2 x} x^2+400 x^4+e^{2 x/3} (40-80 x^2)+e^x (400 x+80 e^{2 x/3} x-800 x^3)} \, dx\) [368]

Optimal result

Integrand size = 117, antiderivative size = 32 \[ \int \frac {-75-150 e^{2 x}-150 x^2+e^{2 x/3} (-15+10 x)+e^x \left (75+5 e^{2 x/3}+300 x\right )}{100+4 e^{4 x/3}-400 x^2+400 e^{2 x} x^2+400 x^4+e^{2 x/3} \left (40-80 x^2\right )+e^x \left (400 x+80 e^{2 x/3} x-800 x^3\right )} \, dx=\frac {3}{4 \left (\frac {5+e^{2 x/3}}{5 \left (e^x-x\right )}+2 x\right )} \] Output:

3/(8*x+4*(exp(2/3*x)+5)/(5*exp(x)-5*x))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 4.05 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.03 \[ \int \frac {-75-150 e^{2 x}-150 x^2+e^{2 x/3} (-15+10 x)+e^x \left (75+5 e^{2 x/3}+300 x\right )}{100+4 e^{4 x/3}-400 x^2+400 e^{2 x} x^2+400 x^4+e^{2 x/3} \left (40-80 x^2\right )+e^x \left (400 x+80 e^{2 x/3} x-800 x^3\right )} \, dx=\frac {15 \left (e^x-x\right )}{4 \left (5+e^{2 x/3}+10 e^x x-10 x^2\right )} \] Input:

Integrate[(-75 - 150*E^(2*x) - 150*x^2 + E^((2*x)/3)*(-15 + 10*x) + E^x*(7 
5 + 5*E^((2*x)/3) + 300*x))/(100 + 4*E^((4*x)/3) - 400*x^2 + 400*E^(2*x)*x 
^2 + 400*x^4 + E^((2*x)/3)*(40 - 80*x^2) + E^x*(400*x + 80*E^((2*x)/3)*x - 
 800*x^3)),x]
 

Output:

(15*(E^x - x))/(4*(5 + E^((2*x)/3) + 10*E^x*x - 10*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.

\(\displaystyle \int \frac {-150 x^2-150 e^{2 x}+e^{2 x/3} (10 x-15)+e^x \left (300 x+5 e^{2 x/3}+75\right )-75}{400 x^4+e^x \left (-800 x^3+80 e^{2 x/3} x+400 x\right )+400 e^{2 x} x^2-400 x^2+e^{2 x/3} \left (40-80 x^2\right )+4 e^{4 x/3}+100} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-150 x^2-150 e^{2 x}+e^{2 x/3} (10 x-15)+e^x \left (300 x+5 e^{2 x/3}+75\right )-75}{4 \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \int -\frac {5 \left (30 x^2+30 e^{2 x}+e^{2 x/3} (3-2 x)-e^x \left (60 x+e^{2 x/3}+15\right )+15\right )}{\left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {5}{4} \int \frac {30 x^2+30 e^{2 x}+e^{2 x/3} (3-2 x)-e^x \left (60 x+e^{2 x/3}+15\right )+15}{\left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -\frac {5}{4} \int \left (\frac {100 e^{2 x/3} x^3+1500 x^3-10 e^{x/3} x^2+600 e^{2 x/3} x^2+3000 x^2-30 e^{x/3} x+x+3}{1000 x^4 \left (10 x^2-10 e^x x-e^{2 x/3}-5\right )}-\frac {3000 e^{2 x/3} x^5+15000 x^5-100 e^{x/3} x^4-3000 e^{2 x/3} x^4-15000 x^4-300 e^{x/3} x^3-2000 e^{2 x/3} x^3-7490 x^3+50 e^{x/3} x^2-3000 e^{2 x/3} x^2-7470 x^2+150 e^{x/3} x-e^{2 x/3} x-5 x-3 e^{2 x/3}-15}{1000 x^4 \left (10 x^2-10 e^x x-e^{2 x/3}-5\right )^2}+\frac {3}{10 x^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {5}{4} \left (15 \int \frac {1}{\left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx+\frac {1}{10} \int \frac {e^{x/3}}{\left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx+3 \int \frac {e^{2 x/3}}{\left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx+\frac {3 \int \frac {e^{2 x/3}}{x^4 \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx}{1000}-\frac {3}{20} \int \frac {e^{x/3}}{x^3 \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx+\frac {\int \frac {e^{2 x/3}}{x^3 \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx}{1000}-\frac {1}{20} \int \frac {e^{x/3}}{x^2 \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx+3 \int \frac {e^{2 x/3}}{x^2 \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx+\frac {3}{10} \int \frac {e^{x/3}}{x \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx+2 \int \frac {e^{2 x/3}}{x \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx-3 \int \frac {e^{2 x/3} x}{\left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )^2}dx+\frac {3}{100} \int \frac {e^{x/3}}{x^3 \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )}dx+\frac {1}{100} \int \frac {e^{x/3}}{x^2 \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )}dx-\frac {3}{5} \int \frac {e^{2 x/3}}{x^2 \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )}dx-\frac {1}{10} \int \frac {e^{2 x/3}}{x \left (-10 x^2+10 e^x x+e^{2 x/3}+5\right )}dx+\frac {3}{200} \int \frac {1}{x^4 \left (10 x^2-10 e^x x-e^{2 x/3}-5\right )^2}dx+\frac {1}{200} \int \frac {1}{x^3 \left (10 x^2-10 e^x x-e^{2 x/3}-5\right )^2}dx+\frac {747}{100} \int \frac {1}{x^2 \left (10 x^2-10 e^x x-e^{2 x/3}-5\right )^2}dx+\frac {749}{100} \int \frac {1}{x \left (10 x^2-10 e^x x-e^{2 x/3}-5\right )^2}dx-15 \int \frac {x}{\left (10 x^2-10 e^x x-e^{2 x/3}-5\right )^2}dx+\frac {3 \int \frac {1}{x^4 \left (10 x^2-10 e^x x-e^{2 x/3}-5\right )}dx}{1000}+\frac {\int \frac {1}{x^3 \left (10 x^2-10 e^x x-e^{2 x/3}-5\right )}dx}{1000}+3 \int \frac {1}{x^2 \left (10 x^2-10 e^x x-e^{2 x/3}-5\right )}dx+\frac {3}{2} \int \frac {1}{x \left (10 x^2-10 e^x x-e^{2 x/3}-5\right )}dx-\frac {3}{10 x}\right )\)

Input:

Int[(-75 - 150*E^(2*x) - 150*x^2 + E^((2*x)/3)*(-15 + 10*x) + E^x*(75 + 5* 
E^((2*x)/3) + 300*x))/(100 + 4*E^((4*x)/3) - 400*x^2 + 400*E^(2*x)*x^2 + 4 
00*x^4 + E^((2*x)/3)*(40 - 80*x^2) + E^x*(400*x + 80*E^((2*x)/3)*x - 800*x 
^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97

method	result	size
parallelrisch	\(-\frac {-15 x +15 \,{\mathrm e}^{x}}{4 \left (-10 \,{\mathrm e}^{x} x +10 x^{2}-{\mathrm e}^{\frac {2 x}{3}}-5\right )}\)	\(31\)
risch	\(\frac {3}{8 x}+\frac {\frac {3 \,{\mathrm e}^{\frac {2 x}{3}}}{8}+\frac {15}{8}}{x \left (-10 \,{\mathrm e}^{x} x +10 x^{2}-{\mathrm e}^{\frac {2 x}{3}}-5\right )}\)	\(38\)

Input:

int((-150*exp(x)^2+(5*exp(2/3*x)+300*x+75)*exp(x)+(10*x-15)*exp(2/3*x)-150 
*x^2-75)/(400*exp(x)^2*x^2+(80*x*exp(2/3*x)-800*x^3+400*x)*exp(x)+4*exp(2/ 
3*x)^2+(-80*x^2+40)*exp(2/3*x)+400*x^4-400*x^2+100),x,method=_RETURNVERBOS 
E)
 

Output:

-1/4*(-15*x+15*exp(x))/(-10*exp(x)*x+10*x^2-exp(2/3*x)-5)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {-75-150 e^{2 x}-150 x^2+e^{2 x/3} (-15+10 x)+e^x \left (75+5 e^{2 x/3}+300 x\right )}{100+4 e^{4 x/3}-400 x^2+400 e^{2 x} x^2+400 x^4+e^{2 x/3} \left (40-80 x^2\right )+e^x \left (400 x+80 e^{2 x/3} x-800 x^3\right )} \, dx=\frac {15 \, {\left (x - e^{x}\right )}}{4 \, {\left (10 \, x^{2} - 10 \, x e^{x} - e^{\left (\frac {2}{3} \, x\right )} - 5\right )}} \] Input:

integrate((-150*exp(x)^2+(5*exp(2/3*x)+300*x+75)*exp(x)+(10*x-15)*exp(2/3* 
x)-150*x^2-75)/(400*exp(x)^2*x^2+(80*x*exp(2/3*x)-800*x^3+400*x)*exp(x)+4* 
exp(2/3*x)^2+(-80*x^2+40)*exp(2/3*x)+400*x^4-400*x^2+100),x, algorithm="fr 
icas")
 

Output:

15/4*(x - e^x)/(10*x^2 - 10*x*e^x - e^(2/3*x) - 5)
 

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.31 \[ \int \frac {-75-150 e^{2 x}-150 x^2+e^{2 x/3} (-15+10 x)+e^x \left (75+5 e^{2 x/3}+300 x\right )}{100+4 e^{4 x/3}-400 x^2+400 e^{2 x} x^2+400 x^4+e^{2 x/3} \left (40-80 x^2\right )+e^x \left (400 x+80 e^{2 x/3} x-800 x^3\right )} \, dx=\frac {- 3 e^{\frac {2 x}{3}} - 15}{- 80 x^{3} + 80 x^{2} e^{x} + 8 x e^{\frac {2 x}{3}} + 40 x} + \frac {3}{8 x} \] Input:

integrate((-150*exp(x)**2+(5*exp(2/3*x)+300*x+75)*exp(x)+(10*x-15)*exp(2/3 
*x)-150*x**2-75)/(400*exp(x)**2*x**2+(80*x*exp(2/3*x)-800*x**3+400*x)*exp( 
x)+4*exp(2/3*x)**2+(-80*x**2+40)*exp(2/3*x)+400*x**4-400*x**2+100),x)
 

Output:

(-3*exp(2*x/3) - 15)/(-80*x**3 + 80*x**2*exp(x) + 8*x*exp(2*x/3) + 40*x) + 
 3/(8*x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88 \[ \int \frac {-75-150 e^{2 x}-150 x^2+e^{2 x/3} (-15+10 x)+e^x \left (75+5 e^{2 x/3}+300 x\right )}{100+4 e^{4 x/3}-400 x^2+400 e^{2 x} x^2+400 x^4+e^{2 x/3} \left (40-80 x^2\right )+e^x \left (400 x+80 e^{2 x/3} x-800 x^3\right )} \, dx=\frac {15 \, {\left (x - e^{x}\right )}}{4 \, {\left (10 \, x^{2} - 10 \, x e^{x} - e^{\left (\frac {2}{3} \, x\right )} - 5\right )}} \] Input:

integrate((-150*exp(x)^2+(5*exp(2/3*x)+300*x+75)*exp(x)+(10*x-15)*exp(2/3* 
x)-150*x^2-75)/(400*exp(x)^2*x^2+(80*x*exp(2/3*x)-800*x^3+400*x)*exp(x)+4* 
exp(2/3*x)^2+(-80*x^2+40)*exp(2/3*x)+400*x^4-400*x^2+100),x, algorithm="ma 
xima")
 

Output:

15/4*(x - e^x)/(10*x^2 - 10*x*e^x - e^(2/3*x) - 5)
 

Giac [A] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.34 \[ \int \frac {-75-150 e^{2 x}-150 x^2+e^{2 x/3} (-15+10 x)+e^x \left (75+5 e^{2 x/3}+300 x\right )}{100+4 e^{4 x/3}-400 x^2+400 e^{2 x} x^2+400 x^4+e^{2 x/3} \left (40-80 x^2\right )+e^x \left (400 x+80 e^{2 x/3} x-800 x^3\right )} \, dx=\frac {3 \, {\left (10 \, x^{2} - 10 \, x e^{x} + e^{\left (\frac {2}{3} \, x\right )} + 5\right )}}{8 \, {\left (10 \, x^{3} - 10 \, x^{2} e^{x} - x e^{\left (\frac {2}{3} \, x\right )} - 5 \, x\right )}} \] Input:

integrate((-150*exp(x)^2+(5*exp(2/3*x)+300*x+75)*exp(x)+(10*x-15)*exp(2/3* 
x)-150*x^2-75)/(400*exp(x)^2*x^2+(80*x*exp(2/3*x)-800*x^3+400*x)*exp(x)+4* 
exp(2/3*x)^2+(-80*x^2+40)*exp(2/3*x)+400*x^4-400*x^2+100),x, algorithm="gi 
ac")
 

Output:

3/8*(10*x^2 - 10*x*e^x + e^(2/3*x) + 5)/(10*x^3 - 10*x^2*e^x - x*e^(2/3*x) 
 - 5*x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-75-150 e^{2 x}-150 x^2+e^{2 x/3} (-15+10 x)+e^x \left (75+5 e^{2 x/3}+300 x\right )}{100+4 e^{4 x/3}-400 x^2+400 e^{2 x} x^2+400 x^4+e^{2 x/3} \left (40-80 x^2\right )+e^x \left (400 x+80 e^{2 x/3} x-800 x^3\right )} \, dx=\int -\frac {150\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (300\,x+5\,{\mathrm {e}}^{\frac {2\,x}{3}}+75\right )-{\mathrm {e}}^{\frac {2\,x}{3}}\,\left (10\,x-15\right )+150\,x^2+75}{4\,{\mathrm {e}}^{\frac {4\,x}{3}}-{\mathrm {e}}^{\frac {2\,x}{3}}\,\left (80\,x^2-40\right )+400\,x^2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^x\,\left (400\,x+80\,x\,{\mathrm {e}}^{\frac {2\,x}{3}}-800\,x^3\right )-400\,x^2+400\,x^4+100} \,d x \] Input:

int(-(150*exp(2*x) - exp(x)*(300*x + 5*exp((2*x)/3) + 75) - exp((2*x)/3)*( 
10*x - 15) + 150*x^2 + 75)/(4*exp((4*x)/3) - exp((2*x)/3)*(80*x^2 - 40) + 
400*x^2*exp(2*x) + exp(x)*(400*x + 80*x*exp((2*x)/3) - 800*x^3) - 400*x^2 
+ 400*x^4 + 100),x)
 

Output:

int(-(150*exp(2*x) - exp(x)*(300*x + 5*exp((2*x)/3) + 75) - exp((2*x)/3)*( 
10*x - 15) + 150*x^2 + 75)/(4*exp((4*x)/3) - exp((2*x)/3)*(80*x^2 - 40) + 
400*x^2*exp(2*x) + exp(x)*(400*x + 80*x*exp((2*x)/3) - 800*x^3) - 400*x^2 
+ 400*x^4 + 100), x)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {-75-150 e^{2 x}-150 x^2+e^{2 x/3} (-15+10 x)+e^x \left (75+5 e^{2 x/3}+300 x\right )}{100+4 e^{4 x/3}-400 x^2+400 e^{2 x} x^2+400 x^4+e^{2 x/3} \left (40-80 x^2\right )+e^x \left (400 x+80 e^{2 x/3} x-800 x^3\right )} \, dx=\frac {15 e^{x}-15 x}{4 e^{\frac {2 x}{3}}+40 e^{x} x -40 x^{2}+20} \] Input:

int((-150*exp(x)^2+(5*exp(2/3*x)+300*x+75)*exp(x)+(10*x-15)*exp(2/3*x)-150 
*x^2-75)/(400*exp(x)^2*x^2+(80*x*exp(2/3*x)-800*x^3+400*x)*exp(x)+4*exp(2/ 
3*x)^2+(-80*x^2+40)*exp(2/3*x)+400*x^4-400*x^2+100),x)
 

Output:

(15*(e**x - x))/(4*(e**((2*x)/3) + 10*e**x*x - 10*x**2 + 5))