\(\int \frac {e^{\frac {-x^3-x^4-2 x^5-x^6}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}} (10 x^3+14 x^4+38 x^5+20 x^6-6 x^7-6 x^8-4 x^9-x^{10}+e^3 (-45 x^2-60 x^3-150 x^4-90 x^5))}{225 e^6+25 x^2-10 x^3-9 x^4-18 x^5-5 x^6+6 x^7+6 x^8+4 x^9+x^{10}+e^3 (-150 x+30 x^2+30 x^3+60 x^4+30 x^5)} \, dx\) [325]

Optimal result

Integrand size = 195, antiderivative size = 30 \[ \int \frac {e^{\frac {-x^3-x^4-2 x^5-x^6}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}} \left (10 x^3+14 x^4+38 x^5+20 x^6-6 x^7-6 x^8-4 x^9-x^{10}+e^3 \left (-45 x^2-60 x^3-150 x^4-90 x^5\right )\right )}{225 e^6+25 x^2-10 x^3-9 x^4-18 x^5-5 x^6+6 x^7+6 x^8+4 x^9+x^{10}+e^3 \left (-150 x+30 x^2+30 x^3+60 x^4+30 x^5\right )} \, dx=e^{\frac {x}{-1+\frac {5-\frac {15 e^3}{x}}{x+\left (x+x^2\right )^2}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {e^{\frac {-x^3-x^4-2 x^5-x^6}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}} \left (10 x^3+14 x^4+38 x^5+20 x^6-6 x^7-6 x^8-4 x^9-x^{10}+e^3 \left (-45 x^2-60 x^3-150 x^4-90 x^5\right )\right )}{225 e^6+25 x^2-10 x^3-9 x^4-18 x^5-5 x^6+6 x^7+6 x^8+4 x^9+x^{10}+e^3 \left (-150 x+30 x^2+30 x^3+60 x^4+30 x^5\right )} \, dx=e^{-\frac {x^3 \left (1+x+2 x^2+x^3\right )}{15 e^3+x \left (-5+x+x^2+2 x^3+x^4\right )}} \] Input:

Integrate[(E^((-x^3 - x^4 - 2*x^5 - x^6)/(15*E^3 - 5*x + x^2 + x^3 + 2*x^4 
 + x^5))*(10*x^3 + 14*x^4 + 38*x^5 + 20*x^6 - 6*x^7 - 6*x^8 - 4*x^9 - x^10 
 + E^3*(-45*x^2 - 60*x^3 - 150*x^4 - 90*x^5)))/(225*E^6 + 25*x^2 - 10*x^3 
- 9*x^4 - 18*x^5 - 5*x^6 + 6*x^7 + 6*x^8 + 4*x^9 + x^10 + E^3*(-150*x + 30 
*x^2 + 30*x^3 + 60*x^4 + 30*x^5)),x]
 

Output:

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

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.

\(\displaystyle \int \frac {\left (-x^{10}-4 x^9-6 x^8-6 x^7+20 x^6+38 x^5+14 x^4+10 x^3+e^3 \left (-90 x^5-150 x^4-60 x^3-45 x^2\right )\right ) \exp \left (\frac {-x^6-2 x^5-x^4-x^3}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right )}{x^{10}+4 x^9+6 x^8+6 x^7-5 x^6-18 x^5-9 x^4-10 x^3+25 x^2+e^3 \left (30 x^5+60 x^4+30 x^3+30 x^2-150 x\right )+225 e^6} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \frac {\left (-x^{10}-4 x^9-6 x^8-6 x^7+20 x^6+38 x^5+14 x^4+10 x^3+e^3 \left (-90 x^5-150 x^4-60 x^3-45 x^2\right )\right ) \exp \left (\frac {-x^6-2 x^5-x^4-x^3}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right )}{\left (x^5+2 x^4+x^3+x^2-5 x+15 e^3\right )^2}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-x^{10}-4 x^9-6 x^8-6 x^7+20 x^6+38 x^5+14 x^4+10 x^3+e^3 \left (-90 x^5-150 x^4-60 x^3-45 x^2\right )\right ) \exp \left (\frac {x^3 \left (-x^3-2 x^2-x-1\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right )}{\left (x^5+2 x^4+x^3+x^2-5 x+15 e^3\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {5 \left (-3 x+12 e^3+2\right ) \exp \left (\frac {x^3 \left (-x^3-2 x^2-x-1\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}-\exp \left (\frac {x^3 \left (-x^3-2 x^2-x-1\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right )+\frac {5 \left (2 \left (1+3 e^3\right ) x^4-\left (1-6 e^3\right ) x^3+\left (22+9 e^3\right ) x^2-5 \left (2+27 e^3\right ) x+15 e^3 \left (2+15 e^3\right )\right ) \exp \left (\frac {x^3 \left (-x^3-2 x^2-x-1\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right )}{\left (x^5+2 x^4+x^3+x^2-5 x+15 e^3\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\int \exp \left (\frac {x^3 \left (-x^3-2 x^2-x-1\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right )dx+75 \left (2+15 e^3\right ) \int \frac {\exp \left (\frac {\left (-x^3-2 x^2-x-1\right ) x^3}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}+3\right )}{\left (x^5+2 x^4+x^3+x^2-5 x+15 e^3\right )^2}dx-25 \left (2+27 e^3\right ) \int \frac {\exp \left (\frac {x^3 \left (-x^3-2 x^2-x-1\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right ) x}{\left (x^5+2 x^4+x^3+x^2-5 x+15 e^3\right )^2}dx+5 \left (22+9 e^3\right ) \int \frac {\exp \left (\frac {x^3 \left (-x^3-2 x^2-x-1\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right ) x^2}{\left (x^5+2 x^4+x^3+x^2-5 x+15 e^3\right )^2}dx-5 \left (1-6 e^3\right ) \int \frac {\exp \left (\frac {x^3 \left (-x^3-2 x^2-x-1\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right ) x^3}{\left (x^5+2 x^4+x^3+x^2-5 x+15 e^3\right )^2}dx+10 \left (1+3 e^3\right ) \int \frac {\exp \left (\frac {x^3 \left (-x^3-2 x^2-x-1\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right ) x^4}{\left (x^5+2 x^4+x^3+x^2-5 x+15 e^3\right )^2}dx-10 \left (1+6 e^3\right ) \int \frac {\exp \left (\frac {x^3 \left (-x^3-2 x^2-x-1\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}dx+15 \int \frac {\exp \left (\frac {x^3 \left (-x^3-2 x^2-x-1\right )}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}\right ) x}{x^5+2 x^4+x^3+x^2-5 x+15 e^3}dx\)

Input:

Int[(E^((-x^3 - x^4 - 2*x^5 - x^6)/(15*E^3 - 5*x + x^2 + x^3 + 2*x^4 + x^5 
))*(10*x^3 + 14*x^4 + 38*x^5 + 20*x^6 - 6*x^7 - 6*x^8 - 4*x^9 - x^10 + E^3 
*(-45*x^2 - 60*x^3 - 150*x^4 - 90*x^5)))/(225*E^6 + 25*x^2 - 10*x^3 - 9*x^ 
4 - 18*x^5 - 5*x^6 + 6*x^7 + 6*x^8 + 4*x^9 + x^10 + E^3*(-150*x + 30*x^2 + 
 30*x^3 + 60*x^4 + 30*x^5)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 2.59 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40

method	result	size
gosper	\({\mathrm e}^{-\frac {x^{3} \left (x^{3}+2 x^{2}+x +1\right )}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}\)	\(42\)
risch	\({\mathrm e}^{-\frac {x^{3} \left (x^{3}+2 x^{2}+x +1\right )}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}\)	\(42\)
parallelrisch	\({\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}\)	\(48\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}+x^{3} {\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}+x^{5} {\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}-5 x \,{\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}+2 x^{4} {\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}+15 \,{\mathrm e}^{3} {\mathrm e}^{\frac {-x^{6}-2 x^{5}-x^{4}-x^{3}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}}}{15 \,{\mathrm e}^{3}+x^{5}+2 x^{4}+x^{3}+x^{2}-5 x}\)	\(333\)

Input:

int(((-90*x^5-150*x^4-60*x^3-45*x^2)*exp(3)-x^10-4*x^9-6*x^8-6*x^7+20*x^6+ 
38*x^5+14*x^4+10*x^3)*exp((-x^6-2*x^5-x^4-x^3)/(15*exp(3)+x^5+2*x^4+x^3+x^ 
2-5*x))/(225*exp(3)^2+(30*x^5+60*x^4+30*x^3+30*x^2-150*x)*exp(3)+x^10+4*x^ 
9+6*x^8+6*x^7-5*x^6-18*x^5-9*x^4-10*x^3+25*x^2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {e^{\frac {-x^3-x^4-2 x^5-x^6}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}} \left (10 x^3+14 x^4+38 x^5+20 x^6-6 x^7-6 x^8-4 x^9-x^{10}+e^3 \left (-45 x^2-60 x^3-150 x^4-90 x^5\right )\right )}{225 e^6+25 x^2-10 x^3-9 x^4-18 x^5-5 x^6+6 x^7+6 x^8+4 x^9+x^{10}+e^3 \left (-150 x+30 x^2+30 x^3+60 x^4+30 x^5\right )} \, dx=e^{\left (-\frac {x^{6} + 2 \, x^{5} + x^{4} + x^{3}}{x^{5} + 2 \, x^{4} + x^{3} + x^{2} - 5 \, x + 15 \, e^{3}}\right )} \] Input:

integrate(((-90*x^5-150*x^4-60*x^3-45*x^2)*exp(3)-x^10-4*x^9-6*x^8-6*x^7+2 
0*x^6+38*x^5+14*x^4+10*x^3)*exp((-x^6-2*x^5-x^4-x^3)/(15*exp(3)+x^5+2*x^4+ 
x^3+x^2-5*x))/(225*exp(3)^2+(30*x^5+60*x^4+30*x^3+30*x^2-150*x)*exp(3)+x^1 
0+4*x^9+6*x^8+6*x^7-5*x^6-18*x^5-9*x^4-10*x^3+25*x^2),x, algorithm="fricas 
")
 

Output:

e^(-(x^6 + 2*x^5 + x^4 + x^3)/(x^5 + 2*x^4 + x^3 + x^2 - 5*x + 15*e^3))
 

Sympy [B] (verification not implemented)

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

Time = 4.84 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {e^{\frac {-x^3-x^4-2 x^5-x^6}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}} \left (10 x^3+14 x^4+38 x^5+20 x^6-6 x^7-6 x^8-4 x^9-x^{10}+e^3 \left (-45 x^2-60 x^3-150 x^4-90 x^5\right )\right )}{225 e^6+25 x^2-10 x^3-9 x^4-18 x^5-5 x^6+6 x^7+6 x^8+4 x^9+x^{10}+e^3 \left (-150 x+30 x^2+30 x^3+60 x^4+30 x^5\right )} \, dx=e^{\frac {- x^{6} - 2 x^{5} - x^{4} - x^{3}}{x^{5} + 2 x^{4} + x^{3} + x^{2} - 5 x + 15 e^{3}}} \] Input:

integrate(((-90*x**5-150*x**4-60*x**3-45*x**2)*exp(3)-x**10-4*x**9-6*x**8- 
6*x**7+20*x**6+38*x**5+14*x**4+10*x**3)*exp((-x**6-2*x**5-x**4-x**3)/(15*e 
xp(3)+x**5+2*x**4+x**3+x**2-5*x))/(225*exp(3)**2+(30*x**5+60*x**4+30*x**3+ 
30*x**2-150*x)*exp(3)+x**10+4*x**9+6*x**8+6*x**7-5*x**6-18*x**5-9*x**4-10* 
x**3+25*x**2),x)
 

Output:

exp((-x**6 - 2*x**5 - x**4 - x**3)/(x**5 + 2*x**4 + x**3 + x**2 - 5*x + 15 
*exp(3)))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (30) = 60\).

Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.10 \[ \int \frac {e^{\frac {-x^3-x^4-2 x^5-x^6}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}} \left (10 x^3+14 x^4+38 x^5+20 x^6-6 x^7-6 x^8-4 x^9-x^{10}+e^3 \left (-45 x^2-60 x^3-150 x^4-90 x^5\right )\right )}{225 e^6+25 x^2-10 x^3-9 x^4-18 x^5-5 x^6+6 x^7+6 x^8+4 x^9+x^{10}+e^3 \left (-150 x+30 x^2+30 x^3+60 x^4+30 x^5\right )} \, dx=e^{\left (-x - \frac {5 \, x^{2}}{x^{5} + 2 \, x^{4} + x^{3} + x^{2} - 5 \, x + 15 \, e^{3}} + \frac {15 \, x e^{3}}{x^{5} + 2 \, x^{4} + x^{3} + x^{2} - 5 \, x + 15 \, e^{3}}\right )} \] Input:

integrate(((-90*x^5-150*x^4-60*x^3-45*x^2)*exp(3)-x^10-4*x^9-6*x^8-6*x^7+2 
0*x^6+38*x^5+14*x^4+10*x^3)*exp((-x^6-2*x^5-x^4-x^3)/(15*exp(3)+x^5+2*x^4+ 
x^3+x^2-5*x))/(225*exp(3)^2+(30*x^5+60*x^4+30*x^3+30*x^2-150*x)*exp(3)+x^1 
0+4*x^9+6*x^8+6*x^7-5*x^6-18*x^5-9*x^4-10*x^3+25*x^2),x, algorithm="maxima 
")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.81 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {e^{\frac {-x^3-x^4-2 x^5-x^6}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}} \left (10 x^3+14 x^4+38 x^5+20 x^6-6 x^7-6 x^8-4 x^9-x^{10}+e^3 \left (-45 x^2-60 x^3-150 x^4-90 x^5\right )\right )}{225 e^6+25 x^2-10 x^3-9 x^4-18 x^5-5 x^6+6 x^7+6 x^8+4 x^9+x^{10}+e^3 \left (-150 x+30 x^2+30 x^3+60 x^4+30 x^5\right )} \, dx=e^{\left (-\frac {x^{6} + 2 \, x^{5} + x^{4} + x^{3}}{x^{5} + 2 \, x^{4} + x^{3} + x^{2} - 5 \, x + 15 \, e^{3}}\right )} \] Input:

integrate(((-90*x^5-150*x^4-60*x^3-45*x^2)*exp(3)-x^10-4*x^9-6*x^8-6*x^7+2 
0*x^6+38*x^5+14*x^4+10*x^3)*exp((-x^6-2*x^5-x^4-x^3)/(15*exp(3)+x^5+2*x^4+ 
x^3+x^2-5*x))/(225*exp(3)^2+(30*x^5+60*x^4+30*x^3+30*x^2-150*x)*exp(3)+x^1 
0+4*x^9+6*x^8+6*x^7-5*x^6-18*x^5-9*x^4-10*x^3+25*x^2),x, algorithm="giac")
 

Output:

e^(-(x^6 + 2*x^5 + x^4 + x^3)/(x^5 + 2*x^4 + x^3 + x^2 - 5*x + 15*e^3))
 

Mupad [B] (verification not implemented)

Time = 3.54 (sec) , antiderivative size = 121, normalized size of antiderivative = 4.03 \[ \int \frac {e^{\frac {-x^3-x^4-2 x^5-x^6}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}} \left (10 x^3+14 x^4+38 x^5+20 x^6-6 x^7-6 x^8-4 x^9-x^{10}+e^3 \left (-45 x^2-60 x^3-150 x^4-90 x^5\right )\right )}{225 e^6+25 x^2-10 x^3-9 x^4-18 x^5-5 x^6+6 x^7+6 x^8+4 x^9+x^{10}+e^3 \left (-150 x+30 x^2+30 x^3+60 x^4+30 x^5\right )} \, dx={\mathrm {e}}^{-\frac {x^3}{x^5+2\,x^4+x^3+x^2-5\,x+15\,{\mathrm {e}}^3}}\,{\mathrm {e}}^{-\frac {x^4}{x^5+2\,x^4+x^3+x^2-5\,x+15\,{\mathrm {e}}^3}}\,{\mathrm {e}}^{-\frac {2\,x^5}{x^5+2\,x^4+x^3+x^2-5\,x+15\,{\mathrm {e}}^3}}\,{\mathrm {e}}^{-\frac {x^6}{x^5+2\,x^4+x^3+x^2-5\,x+15\,{\mathrm {e}}^3}} \] Input:

int(-(exp(-(x^3 + x^4 + 2*x^5 + x^6)/(15*exp(3) - 5*x + x^2 + x^3 + 2*x^4 
+ x^5))*(6*x^7 - 14*x^4 - 38*x^5 - 20*x^6 - 10*x^3 + 6*x^8 + 4*x^9 + x^10 
+ exp(3)*(45*x^2 + 60*x^3 + 150*x^4 + 90*x^5)))/(225*exp(6) + exp(3)*(30*x 
^2 - 150*x + 30*x^3 + 60*x^4 + 30*x^5) + 25*x^2 - 10*x^3 - 9*x^4 - 18*x^5 
- 5*x^6 + 6*x^7 + 6*x^8 + 4*x^9 + x^10),x)
 

Output:

exp(-x^3/(15*exp(3) - 5*x + x^2 + x^3 + 2*x^4 + x^5))*exp(-x^4/(15*exp(3) 
- 5*x + x^2 + x^3 + 2*x^4 + x^5))*exp(-(2*x^5)/(15*exp(3) - 5*x + x^2 + x^ 
3 + 2*x^4 + x^5))*exp(-x^6/(15*exp(3) - 5*x + x^2 + x^3 + 2*x^4 + x^5))
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {-x^3-x^4-2 x^5-x^6}{15 e^3-5 x+x^2+x^3+2 x^4+x^5}} \left (10 x^3+14 x^4+38 x^5+20 x^6-6 x^7-6 x^8-4 x^9-x^{10}+e^3 \left (-45 x^2-60 x^3-150 x^4-90 x^5\right )\right )}{225 e^6+25 x^2-10 x^3-9 x^4-18 x^5-5 x^6+6 x^7+6 x^8+4 x^9+x^{10}+e^3 \left (-150 x+30 x^2+30 x^3+60 x^4+30 x^5\right )} \, dx=\frac {1}{e^{\frac {x^{6}+2 x^{5}+x^{4}+x^{3}}{x^{5}+2 x^{4}+15 e^{3}+x^{3}+x^{2}-5 x}}} \] Input:

int(((-90*x^5-150*x^4-60*x^3-45*x^2)*exp(3)-x^10-4*x^9-6*x^8-6*x^7+20*x^6+ 
38*x^5+14*x^4+10*x^3)*exp((-x^6-2*x^5-x^4-x^3)/(15*exp(3)+x^5+2*x^4+x^3+x^ 
2-5*x))/(225*exp(3)^2+(30*x^5+60*x^4+30*x^3+30*x^2-150*x)*exp(3)+x^10+4*x^ 
9+6*x^8+6*x^7-5*x^6-18*x^5-9*x^4-10*x^3+25*x^2),x)
 

Output:

1/e**((x**6 + 2*x**5 + x**4 + x**3)/(15*e**3 + x**5 + 2*x**4 + x**3 + x**2 
 - 5*x))