\(\int \frac {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} (-300+100 x+2 x^2+50 x^4)}{2500+e^{4 x}+5000 x+2600 x^2+100 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} (100+100 x+2 x^2+50 x^4)} \, dx\) [921]

Optimal result

Integrand size = 125, antiderivative size = 27 \[ \int \frac {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (-300+100 x+2 x^2+50 x^4\right )}{2500+e^{4 x}+5000 x+2600 x^2+100 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (100+100 x+2 x^2+50 x^4\right )} \, dx=x+\frac {8}{2+2 x+x^4+\frac {1}{25} \left (e^{2 x}+x^2\right )} \] Output:

4/(x+1+1/2*x^4+1/50*x^2+1/50*exp(x)^2)+x
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 2.70 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.70 \[ \int \frac {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (-300+100 x+2 x^2+50 x^4\right )}{2500+e^{4 x}+5000 x+2600 x^2+100 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (100+100 x+2 x^2+50 x^4\right )} \, dx=\frac {200+50 x+e^{2 x} x+50 x^2+x^3+25 x^5}{50+e^{2 x}+50 x+x^2+25 x^4} \] Input:

Integrate[(-7500 + E^(4*x) + 4600*x + 2600*x^2 - 19900*x^3 + 2501*x^4 + 25 
00*x^5 + 50*x^6 + 625*x^8 + E^(2*x)*(-300 + 100*x + 2*x^2 + 50*x^4))/(2500 
 + E^(4*x) + 5000*x + 2600*x^2 + 100*x^3 + 2501*x^4 + 2500*x^5 + 50*x^6 + 
625*x^8 + E^(2*x)*(100 + 100*x + 2*x^2 + 50*x^4)),x]
 

Output:

(200 + 50*x + E^(2*x)*x + 50*x^2 + x^3 + 25*x^5)/(50 + E^(2*x) + 50*x + x^ 
2 + 25*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.

\(\displaystyle \int \frac {625 x^8+50 x^6+2500 x^5+2501 x^4-19900 x^3+2600 x^2+e^{2 x} \left (50 x^4+2 x^2+100 x-300\right )+4600 x+e^{4 x}-7500}{625 x^8+50 x^6+2500 x^5+2501 x^4+100 x^3+2600 x^2+e^{2 x} \left (50 x^4+2 x^2+100 x+100\right )+5000 x+e^{4 x}+2500} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {625 x^8+50 x^6+2500 x^5+2501 x^4-19900 x^3+2600 x^2+e^{2 x} \left (50 x^4+2 x^2+100 x-300\right )+4600 x+e^{4 x}-7500}{\left (25 x^4+x^2+50 x+e^{2 x}+50\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {400}{25 x^4+x^2+50 x+e^{2 x}+50}+\frac {400 \left (25 x^4-50 x^3+x^2+49 x+25\right )}{\left (25 x^4+x^2+50 x+e^{2 x}+50\right )^2}+1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 10000 \int \frac {1}{\left (25 x^4+x^2+50 x+e^{2 x}+50\right )^2}dx+19600 \int \frac {x}{\left (25 x^4+x^2+50 x+e^{2 x}+50\right )^2}dx+400 \int \frac {x^2}{\left (25 x^4+x^2+50 x+e^{2 x}+50\right )^2}dx+10000 \int \frac {x^4}{\left (25 x^4+x^2+50 x+e^{2 x}+50\right )^2}dx-400 \int \frac {1}{25 x^4+x^2+50 x+e^{2 x}+50}dx-20000 \int \frac {x^3}{\left (25 x^4+x^2+50 x+e^{2 x}+50\right )^2}dx+x\)

Input:

Int[(-7500 + E^(4*x) + 4600*x + 2600*x^2 - 19900*x^3 + 2501*x^4 + 2500*x^5 
 + 50*x^6 + 625*x^8 + E^(2*x)*(-300 + 100*x + 2*x^2 + 50*x^4))/(2500 + E^( 
4*x) + 5000*x + 2600*x^2 + 100*x^3 + 2501*x^4 + 2500*x^5 + 50*x^6 + 625*x^ 
8 + E^(2*x)*(100 + 100*x + 2*x^2 + 50*x^4)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89

method	result	size
risch	\(x +\frac {200}{25 x^{4}+{\mathrm e}^{2 x}+x^{2}+50 x +50}\)	\(24\)
parallelrisch	\(\frac {25 x^{5}+x^{3}+x \,{\mathrm e}^{2 x}+50 x^{2}+50 x +200}{25 x^{4}+{\mathrm e}^{2 x}+x^{2}+50 x +50}\)	\(45\)

Input:

int((exp(x)^4+(50*x^4+2*x^2+100*x-300)*exp(x)^2+625*x^8+50*x^6+2500*x^5+25 
01*x^4-19900*x^3+2600*x^2+4600*x-7500)/(exp(x)^4+(50*x^4+2*x^2+100*x+100)* 
exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4+100*x^3+2600*x^2+5000*x+2500),x, 
method=_RETURNVERBOSE)
 

Output:

x+200/(25*x^4+exp(2*x)+x^2+50*x+50)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (-300+100 x+2 x^2+50 x^4\right )}{2500+e^{4 x}+5000 x+2600 x^2+100 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (100+100 x+2 x^2+50 x^4\right )} \, dx=\frac {25 \, x^{5} + x^{3} + 50 \, x^{2} + x e^{\left (2 \, x\right )} + 50 \, x + 200}{25 \, x^{4} + x^{2} + 50 \, x + e^{\left (2 \, x\right )} + 50} \] Input:

integrate((exp(x)^4+(50*x^4+2*x^2+100*x-300)*exp(x)^2+625*x^8+50*x^6+2500* 
x^5+2501*x^4-19900*x^3+2600*x^2+4600*x-7500)/(exp(x)^4+(50*x^4+2*x^2+100*x 
+100)*exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4+100*x^3+2600*x^2+5000*x+25 
00),x, algorithm="fricas")
 

Output:

(25*x^5 + x^3 + 50*x^2 + x*e^(2*x) + 50*x + 200)/(25*x^4 + x^2 + 50*x + e^ 
(2*x) + 50)
 

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (-300+100 x+2 x^2+50 x^4\right )}{2500+e^{4 x}+5000 x+2600 x^2+100 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (100+100 x+2 x^2+50 x^4\right )} \, dx=x + \frac {200}{25 x^{4} + x^{2} + 50 x + e^{2 x} + 50} \] Input:

integrate((exp(x)**4+(50*x**4+2*x**2+100*x-300)*exp(x)**2+625*x**8+50*x**6 
+2500*x**5+2501*x**4-19900*x**3+2600*x**2+4600*x-7500)/(exp(x)**4+(50*x**4 
+2*x**2+100*x+100)*exp(x)**2+625*x**8+50*x**6+2500*x**5+2501*x**4+100*x**3 
+2600*x**2+5000*x+2500),x)
 

Output:

x + 200/(25*x**4 + x**2 + 50*x + exp(2*x) + 50)
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (-300+100 x+2 x^2+50 x^4\right )}{2500+e^{4 x}+5000 x+2600 x^2+100 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (100+100 x+2 x^2+50 x^4\right )} \, dx=\frac {25 \, x^{5} + x^{3} + 50 \, x^{2} + x e^{\left (2 \, x\right )} + 50 \, x + 200}{25 \, x^{4} + x^{2} + 50 \, x + e^{\left (2 \, x\right )} + 50} \] Input:

integrate((exp(x)^4+(50*x^4+2*x^2+100*x-300)*exp(x)^2+625*x^8+50*x^6+2500* 
x^5+2501*x^4-19900*x^3+2600*x^2+4600*x-7500)/(exp(x)^4+(50*x^4+2*x^2+100*x 
+100)*exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4+100*x^3+2600*x^2+5000*x+25 
00),x, algorithm="maxima")
 

Output:

(25*x^5 + x^3 + 50*x^2 + x*e^(2*x) + 50*x + 200)/(25*x^4 + x^2 + 50*x + e^ 
(2*x) + 50)
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.63 \[ \int \frac {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (-300+100 x+2 x^2+50 x^4\right )}{2500+e^{4 x}+5000 x+2600 x^2+100 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (100+100 x+2 x^2+50 x^4\right )} \, dx=\frac {25 \, x^{5} + x^{3} + 50 \, x^{2} + x e^{\left (2 \, x\right )} + 50 \, x + 200}{25 \, x^{4} + x^{2} + 50 \, x + e^{\left (2 \, x\right )} + 50} \] Input:

integrate((exp(x)^4+(50*x^4+2*x^2+100*x-300)*exp(x)^2+625*x^8+50*x^6+2500* 
x^5+2501*x^4-19900*x^3+2600*x^2+4600*x-7500)/(exp(x)^4+(50*x^4+2*x^2+100*x 
+100)*exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4+100*x^3+2600*x^2+5000*x+25 
00),x, algorithm="giac")
 

Output:

(25*x^5 + x^3 + 50*x^2 + x*e^(2*x) + 50*x + 200)/(25*x^4 + x^2 + 50*x + e^ 
(2*x) + 50)
 

Mupad [B] (verification not implemented)

Time = 8.21 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (-300+100 x+2 x^2+50 x^4\right )}{2500+e^{4 x}+5000 x+2600 x^2+100 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (100+100 x+2 x^2+50 x^4\right )} \, dx=x+\frac {200}{50\,x+{\mathrm {e}}^{2\,x}+x^2+25\,x^4+50} \] Input:

int((4600*x + exp(4*x) + exp(2*x)*(100*x + 2*x^2 + 50*x^4 - 300) + 2600*x^ 
2 - 19900*x^3 + 2501*x^4 + 2500*x^5 + 50*x^6 + 625*x^8 - 7500)/(5000*x + e 
xp(4*x) + exp(2*x)*(100*x + 2*x^2 + 50*x^4 + 100) + 2600*x^2 + 100*x^3 + 2 
501*x^4 + 2500*x^5 + 50*x^6 + 625*x^8 + 2500),x)
 

Output:

x + 200/(50*x + exp(2*x) + x^2 + 25*x^4 + 50)
 

Reduce [F]

\[ \int \frac {-7500+e^{4 x}+4600 x+2600 x^2-19900 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (-300+100 x+2 x^2+50 x^4\right )}{2500+e^{4 x}+5000 x+2600 x^2+100 x^3+2501 x^4+2500 x^5+50 x^6+625 x^8+e^{2 x} \left (100+100 x+2 x^2+50 x^4\right )} \, dx =\text {Too large to display} \] Input:

int((exp(x)^4+(50*x^4+2*x^2+100*x-300)*exp(x)^2+625*x^8+50*x^6+2500*x^5+25 
01*x^4-19900*x^3+2600*x^2+4600*x-7500)/(exp(x)^4+(50*x^4+2*x^2+100*x+100)* 
exp(x)^2+625*x^8+50*x^6+2500*x^5+2501*x^4+100*x^3+2600*x^2+5000*x+2500),x)
 

Output:

int(e**(4*x)/(e**(4*x) + 50*e**(2*x)*x**4 + 2*e**(2*x)*x**2 + 100*e**(2*x) 
*x + 100*e**(2*x) + 625*x**8 + 50*x**6 + 2500*x**5 + 2501*x**4 + 100*x**3 
+ 2600*x**2 + 5000*x + 2500),x) - 300*int(e**(2*x)/(e**(4*x) + 50*e**(2*x) 
*x**4 + 2*e**(2*x)*x**2 + 100*e**(2*x)*x + 100*e**(2*x) + 625*x**8 + 50*x* 
*6 + 2500*x**5 + 2501*x**4 + 100*x**3 + 2600*x**2 + 5000*x + 2500),x) + 62 
5*int(x**8/(e**(4*x) + 50*e**(2*x)*x**4 + 2*e**(2*x)*x**2 + 100*e**(2*x)*x 
 + 100*e**(2*x) + 625*x**8 + 50*x**6 + 2500*x**5 + 2501*x**4 + 100*x**3 + 
2600*x**2 + 5000*x + 2500),x) + 50*int(x**6/(e**(4*x) + 50*e**(2*x)*x**4 + 
 2*e**(2*x)*x**2 + 100*e**(2*x)*x + 100*e**(2*x) + 625*x**8 + 50*x**6 + 25 
00*x**5 + 2501*x**4 + 100*x**3 + 2600*x**2 + 5000*x + 2500),x) + 2500*int( 
x**5/(e**(4*x) + 50*e**(2*x)*x**4 + 2*e**(2*x)*x**2 + 100*e**(2*x)*x + 100 
*e**(2*x) + 625*x**8 + 50*x**6 + 2500*x**5 + 2501*x**4 + 100*x**3 + 2600*x 
**2 + 5000*x + 2500),x) + 2501*int(x**4/(e**(4*x) + 50*e**(2*x)*x**4 + 2*e 
**(2*x)*x**2 + 100*e**(2*x)*x + 100*e**(2*x) + 625*x**8 + 50*x**6 + 2500*x 
**5 + 2501*x**4 + 100*x**3 + 2600*x**2 + 5000*x + 2500),x) - 19900*int(x** 
3/(e**(4*x) + 50*e**(2*x)*x**4 + 2*e**(2*x)*x**2 + 100*e**(2*x)*x + 100*e* 
*(2*x) + 625*x**8 + 50*x**6 + 2500*x**5 + 2501*x**4 + 100*x**3 + 2600*x**2 
 + 5000*x + 2500),x) + 2600*int(x**2/(e**(4*x) + 50*e**(2*x)*x**4 + 2*e**( 
2*x)*x**2 + 100*e**(2*x)*x + 100*e**(2*x) + 625*x**8 + 50*x**6 + 2500*x**5 
 + 2501*x**4 + 100*x**3 + 2600*x**2 + 5000*x + 2500),x) + 50*int((e**(2...