\(\int \frac {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} (9-4 x+2 x^2+e^{-4+x} (8-2 x+2 x^2))}{625+e^{4 e^{-4+x}+4 x}+1250 x^2+625 x^4+e^{2 e^{-4+x}+2 x} (50+50 x^2)} \, dx\) [2270]

Optimal result

Integrand size = 99, antiderivative size = 31 \[ \int \frac {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} \left (9-4 x+2 x^2+e^{-4+x} \left (8-2 x+2 x^2\right )\right )}{625+e^{4 e^{-4+x}+4 x}+1250 x^2+625 x^4+e^{2 e^{-4+x}+2 x} \left (50+50 x^2\right )} \, dx=\frac {-4+x-x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \] Output:

(-x^2+x-4)/(25*x^2+25+exp(x+exp(-4+x))^2)
 

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.03 \[ \int \frac {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} \left (9-4 x+2 x^2+e^{-4+x} \left (8-2 x+2 x^2\right )\right )}{625+e^{4 e^{-4+x}+4 x}+1250 x^2+625 x^4+e^{2 e^{-4+x}+2 x} \left (50+50 x^2\right )} \, dx=-\frac {4-x+x^2}{25+e^{2 e^{-4+x}+2 x}+25 x^2} \] Input:

Integrate[(25 + 150*x - 25*x^2 + E^(2*E^(-4 + x) + 2*x)*(9 - 4*x + 2*x^2 + 
 E^(-4 + x)*(8 - 2*x + 2*x^2)))/(625 + E^(4*E^(-4 + x) + 4*x) + 1250*x^2 + 
 625*x^4 + E^(2*E^(-4 + x) + 2*x)*(50 + 50*x^2)),x]
 

Output:

-((4 - x + x^2)/(25 + E^(2*E^(-4 + x) + 2*x) + 25*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.

\(\displaystyle \int \frac {-25 x^2+e^{2 x+2 e^{x-4}} \left (2 x^2+e^{x-4} \left (2 x^2-2 x+8\right )-4 x+9\right )+150 x+25}{625 x^4+1250 x^2+e^{2 x+2 e^{x-4}} \left (50 x^2+50\right )+e^{4 x+4 e^{x-4}}+625} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-25 x^2+e^{2 x+2 e^{x-4}} \left (2 x^2+e^{x-4} \left (2 x^2-2 x+8\right )-4 x+9\right )+150 x+25}{\left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 e^x x^2+2 e^4 x^2-2 e^x x-4 e^4 x+8 e^x+9 e^4}{e^4 \left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )}-\frac {50 \left (x^2-x+4\right ) \left (e^x x^2+e^4 x^2-e^4 x+e^x+e^4\right )}{e^4 \left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -200 \int \frac {1}{\left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}dx-\frac {200 \int \frac {e^x}{\left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}dx}{e^4}+250 \int \frac {x}{\left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}dx+\frac {50 \int \frac {e^x x}{\left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}dx}{e^4}-300 \int \frac {x^2}{\left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}dx-\frac {250 \int \frac {e^x x^2}{\left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}dx}{e^4}+9 \int \frac {1}{25 x^2+e^{2 x+2 e^{x-4}}+25}dx+\frac {8 \int \frac {e^x}{25 x^2+e^{2 x+2 e^{x-4}}+25}dx}{e^4}-4 \int \frac {x}{25 x^2+e^{2 x+2 e^{x-4}}+25}dx-\frac {2 \int \frac {e^x x}{25 x^2+e^{2 x+2 e^{x-4}}+25}dx}{e^4}+2 \int \frac {x^2}{25 x^2+e^{2 x+2 e^{x-4}}+25}dx+\frac {2 \int \frac {e^x x^2}{25 x^2+e^{2 x+2 e^{x-4}}+25}dx}{e^4}-50 \int \frac {x^4}{\left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}dx-\frac {50 \int \frac {e^x x^4}{\left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}dx}{e^4}+100 \int \frac {x^3}{\left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}dx+\frac {50 \int \frac {e^x x^3}{\left (25 x^2+e^{2 x+2 e^{x-4}}+25\right )^2}dx}{e^4}\)

Input:

Int[(25 + 150*x - 25*x^2 + E^(2*E^(-4 + x) + 2*x)*(9 - 4*x + 2*x^2 + E^(-4 
 + x)*(8 - 2*x + 2*x^2)))/(625 + E^(4*E^(-4 + x) + 4*x) + 1250*x^2 + 625*x 
^4 + E^(2*E^(-4 + x) + 2*x)*(50 + 50*x^2)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90

method	result	size
parallelrisch	\(\frac {-x^{2}+x -4}{25 x^{2}+25+{\mathrm e}^{2 x +2 \,{\mathrm e}^{x -4}}}\)	\(28\)
risch	\(-\frac {x^{2}-x +4}{25 x^{2}+25+{\mathrm e}^{2 x +2 \,{\mathrm e}^{x -4}}}\)	\(31\)

Input:

int((((2*x^2-2*x+8)*exp(x-4)+2*x^2-4*x+9)*exp(x+exp(x-4))^2-25*x^2+150*x+2 
5)/(exp(x+exp(x-4))^4+(50*x^2+50)*exp(x+exp(x-4))^2+625*x^4+1250*x^2+625), 
x,method=_RETURNVERBOSE)
 

Output:

(-x^2+x-4)/(25*x^2+25+exp(x+exp(x-4))^2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} \left (9-4 x+2 x^2+e^{-4+x} \left (8-2 x+2 x^2\right )\right )}{625+e^{4 e^{-4+x}+4 x}+1250 x^2+625 x^4+e^{2 e^{-4+x}+2 x} \left (50+50 x^2\right )} \, dx=-\frac {x^{2} - x + 4}{25 \, x^{2} + e^{\left (2 \, x + 2 \, e^{\left (x - 4\right )}\right )} + 25} \] Input:

integrate((((2*x^2-2*x+8)*exp(-4+x)+2*x^2-4*x+9)*exp(x+exp(-4+x))^2-25*x^2 
+150*x+25)/(exp(x+exp(-4+x))^4+(50*x^2+50)*exp(x+exp(-4+x))^2+625*x^4+1250 
*x^2+625),x, algorithm="fricas")
 

Output:

-(x^2 - x + 4)/(25*x^2 + e^(2*x + 2*e^(x - 4)) + 25)
 

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} \left (9-4 x+2 x^2+e^{-4+x} \left (8-2 x+2 x^2\right )\right )}{625+e^{4 e^{-4+x}+4 x}+1250 x^2+625 x^4+e^{2 e^{-4+x}+2 x} \left (50+50 x^2\right )} \, dx=\frac {- x^{2} + x - 4}{25 x^{2} + e^{2 x + 2 e^{x - 4}} + 25} \] Input:

integrate((((2*x**2-2*x+8)*exp(-4+x)+2*x**2-4*x+9)*exp(x+exp(-4+x))**2-25* 
x**2+150*x+25)/(exp(x+exp(-4+x))**4+(50*x**2+50)*exp(x+exp(-4+x))**2+625*x 
**4+1250*x**2+625),x)
 

Output:

(-x**2 + x - 4)/(25*x**2 + exp(2*x + 2*exp(x - 4)) + 25)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} \left (9-4 x+2 x^2+e^{-4+x} \left (8-2 x+2 x^2\right )\right )}{625+e^{4 e^{-4+x}+4 x}+1250 x^2+625 x^4+e^{2 e^{-4+x}+2 x} \left (50+50 x^2\right )} \, dx=-\frac {x^{2} - x + 4}{25 \, x^{2} + e^{\left (2 \, x + 2 \, e^{\left (x - 4\right )}\right )} + 25} \] Input:

integrate((((2*x^2-2*x+8)*exp(-4+x)+2*x^2-4*x+9)*exp(x+exp(-4+x))^2-25*x^2 
+150*x+25)/(exp(x+exp(-4+x))^4+(50*x^2+50)*exp(x+exp(-4+x))^2+625*x^4+1250 
*x^2+625),x, algorithm="maxima")
 

Output:

-(x^2 - x + 4)/(25*x^2 + e^(2*x + 2*e^(x - 4)) + 25)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} \left (9-4 x+2 x^2+e^{-4+x} \left (8-2 x+2 x^2\right )\right )}{625+e^{4 e^{-4+x}+4 x}+1250 x^2+625 x^4+e^{2 e^{-4+x}+2 x} \left (50+50 x^2\right )} \, dx=-\frac {x^{2} - x + 4}{25 \, x^{2} + e^{\left (2 \, x + 2 \, e^{\left (x - 4\right )}\right )} + 25} \] Input:

integrate((((2*x^2-2*x+8)*exp(-4+x)+2*x^2-4*x+9)*exp(x+exp(-4+x))^2-25*x^2 
+150*x+25)/(exp(x+exp(-4+x))^4+(50*x^2+50)*exp(x+exp(-4+x))^2+625*x^4+1250 
*x^2+625),x, algorithm="giac")
 

Output:

-(x^2 - x + 4)/(25*x^2 + e^(2*x + 2*e^(x - 4)) + 25)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} \left (9-4 x+2 x^2+e^{-4+x} \left (8-2 x+2 x^2\right )\right )}{625+e^{4 e^{-4+x}+4 x}+1250 x^2+625 x^4+e^{2 e^{-4+x}+2 x} \left (50+50 x^2\right )} \, dx=\int \frac {150\,x+{\mathrm {e}}^{2\,x+2\,{\mathrm {e}}^{x-4}}\,\left ({\mathrm {e}}^{x-4}\,\left (2\,x^2-2\,x+8\right )-4\,x+2\,x^2+9\right )-25\,x^2+25}{{\mathrm {e}}^{4\,x+4\,{\mathrm {e}}^{x-4}}+{\mathrm {e}}^{2\,x+2\,{\mathrm {e}}^{x-4}}\,\left (50\,x^2+50\right )+1250\,x^2+625\,x^4+625} \,d x \] Input:

int((150*x + exp(2*x + 2*exp(x - 4))*(exp(x - 4)*(2*x^2 - 2*x + 8) - 4*x + 
 2*x^2 + 9) - 25*x^2 + 25)/(exp(4*x + 4*exp(x - 4)) + exp(2*x + 2*exp(x - 
4))*(50*x^2 + 50) + 1250*x^2 + 625*x^4 + 625),x)
 

Output:

int((150*x + exp(2*x + 2*exp(x - 4))*(exp(x - 4)*(2*x^2 - 2*x + 8) - 4*x + 
 2*x^2 + 9) - 25*x^2 + 25)/(exp(4*x + 4*exp(x - 4)) + exp(2*x + 2*exp(x - 
4))*(50*x^2 + 50) + 1250*x^2 + 625*x^4 + 625), x)
 

Reduce [F]

\[ \int \frac {25+150 x-25 x^2+e^{2 e^{-4+x}+2 x} \left (9-4 x+2 x^2+e^{-4+x} \left (8-2 x+2 x^2\right )\right )}{625+e^{4 e^{-4+x}+4 x}+1250 x^2+625 x^4+e^{2 e^{-4+x}+2 x} \left (50+50 x^2\right )} \, dx=\frac {8 \left (\int \frac {e^{\frac {2 e^{x}+3 e^{4} x}{e^{4}}}}{e^{\frac {4 e^{x}+4 e^{4} x}{e^{4}}}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}} x^{2}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}}+625 x^{4}+1250 x^{2}+625}d x \right )+9 \left (\int \frac {e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}}}{e^{\frac {4 e^{x}+4 e^{4} x}{e^{4}}}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}} x^{2}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}}+625 x^{4}+1250 x^{2}+625}d x \right ) e^{4}-25 \left (\int \frac {x^{2}}{e^{\frac {4 e^{x}+4 e^{4} x}{e^{4}}}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}} x^{2}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}}+625 x^{4}+1250 x^{2}+625}d x \right ) e^{4}+2 \left (\int \frac {e^{\frac {2 e^{x}+3 e^{4} x}{e^{4}}} x^{2}}{e^{\frac {4 e^{x}+4 e^{4} x}{e^{4}}}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}} x^{2}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}}+625 x^{4}+1250 x^{2}+625}d x \right )-2 \left (\int \frac {e^{\frac {2 e^{x}+3 e^{4} x}{e^{4}}} x}{e^{\frac {4 e^{x}+4 e^{4} x}{e^{4}}}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}} x^{2}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}}+625 x^{4}+1250 x^{2}+625}d x \right )+2 \left (\int \frac {e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}} x^{2}}{e^{\frac {4 e^{x}+4 e^{4} x}{e^{4}}}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}} x^{2}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}}+625 x^{4}+1250 x^{2}+625}d x \right ) e^{4}-4 \left (\int \frac {e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}} x}{e^{\frac {4 e^{x}+4 e^{4} x}{e^{4}}}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}} x^{2}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}}+625 x^{4}+1250 x^{2}+625}d x \right ) e^{4}+150 \left (\int \frac {x}{e^{\frac {4 e^{x}+4 e^{4} x}{e^{4}}}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}} x^{2}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}}+625 x^{4}+1250 x^{2}+625}d x \right ) e^{4}+25 \left (\int \frac {1}{e^{\frac {4 e^{x}+4 e^{4} x}{e^{4}}}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}} x^{2}+50 e^{\frac {2 e^{x}+2 e^{4} x}{e^{4}}}+625 x^{4}+1250 x^{2}+625}d x \right ) e^{4}}{e^{4}} \] Input:

int((((2*x^2-2*x+8)*exp(-4+x)+2*x^2-4*x+9)*exp(x+exp(-4+x))^2-25*x^2+150*x 
+25)/(exp(x+exp(-4+x))^4+(50*x^2+50)*exp(x+exp(-4+x))^2+625*x^4+1250*x^2+6 
25),x)
 

Output:

(8*int(e**((2*e**x + 3*e**4*x)/e**4)/(e**((4*e**x + 4*e**4*x)/e**4) + 50*e 
**((2*e**x + 2*e**4*x)/e**4)*x**2 + 50*e**((2*e**x + 2*e**4*x)/e**4) + 625 
*x**4 + 1250*x**2 + 625),x) + 9*int(e**((2*e**x + 2*e**4*x)/e**4)/(e**((4* 
e**x + 4*e**4*x)/e**4) + 50*e**((2*e**x + 2*e**4*x)/e**4)*x**2 + 50*e**((2 
*e**x + 2*e**4*x)/e**4) + 625*x**4 + 1250*x**2 + 625),x)*e**4 - 25*int(x** 
2/(e**((4*e**x + 4*e**4*x)/e**4) + 50*e**((2*e**x + 2*e**4*x)/e**4)*x**2 + 
 50*e**((2*e**x + 2*e**4*x)/e**4) + 625*x**4 + 1250*x**2 + 625),x)*e**4 + 
2*int((e**((2*e**x + 3*e**4*x)/e**4)*x**2)/(e**((4*e**x + 4*e**4*x)/e**4) 
+ 50*e**((2*e**x + 2*e**4*x)/e**4)*x**2 + 50*e**((2*e**x + 2*e**4*x)/e**4) 
 + 625*x**4 + 1250*x**2 + 625),x) - 2*int((e**((2*e**x + 3*e**4*x)/e**4)*x 
)/(e**((4*e**x + 4*e**4*x)/e**4) + 50*e**((2*e**x + 2*e**4*x)/e**4)*x**2 + 
 50*e**((2*e**x + 2*e**4*x)/e**4) + 625*x**4 + 1250*x**2 + 625),x) + 2*int 
((e**((2*e**x + 2*e**4*x)/e**4)*x**2)/(e**((4*e**x + 4*e**4*x)/e**4) + 50* 
e**((2*e**x + 2*e**4*x)/e**4)*x**2 + 50*e**((2*e**x + 2*e**4*x)/e**4) + 62 
5*x**4 + 1250*x**2 + 625),x)*e**4 - 4*int((e**((2*e**x + 2*e**4*x)/e**4)*x 
)/(e**((4*e**x + 4*e**4*x)/e**4) + 50*e**((2*e**x + 2*e**4*x)/e**4)*x**2 + 
 50*e**((2*e**x + 2*e**4*x)/e**4) + 625*x**4 + 1250*x**2 + 625),x)*e**4 + 
150*int(x/(e**((4*e**x + 4*e**4*x)/e**4) + 50*e**((2*e**x + 2*e**4*x)/e**4 
)*x**2 + 50*e**((2*e**x + 2*e**4*x)/e**4) + 625*x**4 + 1250*x**2 + 625),x) 
*e**4 + 25*int(1/(e**((4*e**x + 4*e**4*x)/e**4) + 50*e**((2*e**x + 2*e*...