\(\int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x (-105+430 x-645 x^2+430 x^3-110 x^4)}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} (1-4 x+6 x^2-4 x^3+x^4)+e^x (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6)} \, dx\) [1498]

Optimal result

Integrand size = 145, antiderivative size = 30 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\frac {5}{-x+x^2+\frac {e^x}{21+\frac {x^2}{(1-x)^2}}} \] Output:

5/(x^2-x+exp(x)/(x^2/(1-x)^2+21))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 3.88 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=-\frac {5 \left (-21+42 x-22 x^2\right )}{(-1+x) \left (e^x (-1+x)+x \left (21-42 x+22 x^2\right )\right )} \] Input:

Integrate[(2205 - 13230*x + 31080*x^2 - 36120*x^3 + 20900*x^4 - 4840*x^5 + 
 E^x*(-105 + 430*x - 645*x^2 + 430*x^3 - 110*x^4))/(441*x^2 - 2646*x^3 + 6 
657*x^4 - 8988*x^5 + 6868*x^6 - 2816*x^7 + 484*x^8 + E^(2*x)*(1 - 4*x + 6* 
x^2 - 4*x^3 + x^4) + E^x*(-42*x + 210*x^2 - 422*x^3 + 426*x^4 - 216*x^5 + 
44*x^6)),x]
 

Output:

(-5*(-21 + 42*x - 22*x^2))/((-1 + x)*(E^x*(-1 + x) + x*(21 - 42*x + 22*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.

\(\displaystyle \int \frac {-4840 x^5+20900 x^4-36120 x^3+31080 x^2+e^x \left (-110 x^4+430 x^3-645 x^2+430 x-105\right )-13230 x+2205}{484 x^8-2816 x^7+6868 x^6-8988 x^5+6657 x^4-2646 x^3+441 x^2+e^{2 x} \left (x^4-4 x^3+6 x^2-4 x+1\right )+e^x \left (44 x^6-216 x^5+426 x^4-422 x^3+210 x^2-42 x\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {5 \left (-\left ((2 x-1) \left (22 x^2-42 x+21\right )^2\right )-e^x \left (22 x^4-86 x^3+129 x^2-86 x+21\right )\right )}{(1-x)^2 \left (x \left (22 x^2-42 x+21\right )+e^x (x-1)\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 5 \int \frac {(1-2 x) \left (22 x^2-42 x+21\right )^2-e^x \left (22 x^4-86 x^3+129 x^2-86 x+21\right )}{(1-x)^2 \left (e^x (1-x)-x \left (22 x^2-42 x+21\right )\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 5 \int \left (\frac {484 x^6-3300 x^5+8760 x^4-11760 x^3+8463 x^2-3087 x+441}{(x-1)^2 \left (22 x^3-42 x^2+e^x x+21 x-e^x\right )^2}-\frac {22 x^3-64 x^2+65 x-21}{(x-1)^2 \left (22 x^3-42 x^2+e^x x+21 x-e^x\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 5 \left (443 \int \frac {1}{\left (22 x^3-42 x^2+e^x x+21 x-e^x\right )^2}dx+\int \frac {1}{(x-1)^2 \left (22 x^3-42 x^2+e^x x+21 x-e^x\right )^2}dx+3 \int \frac {1}{(x-1) \left (22 x^3-42 x^2+e^x x+21 x-e^x\right )^2}dx-2204 \int \frac {x}{\left (22 x^3-42 x^2+e^x x+21 x-e^x\right )^2}dx+3612 \int \frac {x^2}{\left (22 x^3-42 x^2+e^x x+21 x-e^x\right )^2}dx-2332 \int \frac {x^3}{\left (22 x^3-42 x^2+e^x x+21 x-e^x\right )^2}dx+20 \int \frac {1}{22 x^3-42 x^2+e^x x+21 x-e^x}dx-2 \int \frac {1}{(x-1)^2 \left (22 x^3-42 x^2+e^x x+21 x-e^x\right )}dx-3 \int \frac {1}{(x-1) \left (22 x^3-42 x^2+e^x x+21 x-e^x\right )}dx-22 \int \frac {x}{22 x^3-42 x^2+e^x x+21 x-e^x}dx+484 \int \frac {x^4}{\left (22 x^3-42 x^2+e^x x+21 x-e^x\right )^2}dx\right )\)

Input:

Int[(2205 - 13230*x + 31080*x^2 - 36120*x^3 + 20900*x^4 - 4840*x^5 + E^x*( 
-105 + 430*x - 645*x^2 + 430*x^3 - 110*x^4))/(441*x^2 - 2646*x^3 + 6657*x^ 
4 - 8988*x^5 + 6868*x^6 - 2816*x^7 + 484*x^8 + E^(2*x)*(1 - 4*x + 6*x^2 - 
4*x^3 + x^4) + E^x*(-42*x + 210*x^2 - 422*x^3 + 426*x^4 - 216*x^5 + 44*x^6 
)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40

method	result	size
risch	\(\frac {110 x^{2}-210 x +105}{\left (-1+x \right ) \left (22 x^{3}-42 x^{2}+{\mathrm e}^{x} x +21 x -{\mathrm e}^{x}\right )}\)	\(42\)
norman	\(\frac {110 x^{2}-210 x +105}{22 x^{4}+{\mathrm e}^{x} x^{2}-64 x^{3}-2 \,{\mathrm e}^{x} x +63 x^{2}+{\mathrm e}^{x}-21 x}\)	\(46\)
parallelrisch	\(\frac {110 x^{2}-210 x +105}{22 x^{4}+{\mathrm e}^{x} x^{2}-64 x^{3}-2 \,{\mathrm e}^{x} x +63 x^{2}+{\mathrm e}^{x}-21 x}\)	\(46\)

Input:

int(((-110*x^4+430*x^3-645*x^2+430*x-105)*exp(x)-4840*x^5+20900*x^4-36120* 
x^3+31080*x^2-13230*x+2205)/((x^4-4*x^3+6*x^2-4*x+1)*exp(x)^2+(44*x^6-216* 
x^5+426*x^4-422*x^3+210*x^2-42*x)*exp(x)+484*x^8-2816*x^7+6868*x^6-8988*x^ 
5+6657*x^4-2646*x^3+441*x^2),x,method=_RETURNVERBOSE)
 

Output:

5*(22*x^2-42*x+21)/(-1+x)/(22*x^3-42*x^2+exp(x)*x+21*x-exp(x))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\frac {5 \, {\left (22 \, x^{2} - 42 \, x + 21\right )}}{22 \, x^{4} - 64 \, x^{3} + 63 \, x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{x} - 21 \, x} \] Input:

integrate(((-110*x^4+430*x^3-645*x^2+430*x-105)*exp(x)-4840*x^5+20900*x^4- 
36120*x^3+31080*x^2-13230*x+2205)/((x^4-4*x^3+6*x^2-4*x+1)*exp(x)^2+(44*x^ 
6-216*x^5+426*x^4-422*x^3+210*x^2-42*x)*exp(x)+484*x^8-2816*x^7+6868*x^6-8 
988*x^5+6657*x^4-2646*x^3+441*x^2),x, algorithm="fricas")
 

Output:

5*(22*x^2 - 42*x + 21)/(22*x^4 - 64*x^3 + 63*x^2 + (x^2 - 2*x + 1)*e^x - 2 
1*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (19) = 38\).

Time = 0.15 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\frac {110 x^{2} - 210 x + 105}{22 x^{4} - 64 x^{3} + 63 x^{2} - 21 x + \left (x^{2} - 2 x + 1\right ) e^{x}} \] Input:

integrate(((-110*x**4+430*x**3-645*x**2+430*x-105)*exp(x)-4840*x**5+20900* 
x**4-36120*x**3+31080*x**2-13230*x+2205)/((x**4-4*x**3+6*x**2-4*x+1)*exp(x 
)**2+(44*x**6-216*x**5+426*x**4-422*x**3+210*x**2-42*x)*exp(x)+484*x**8-28 
16*x**7+6868*x**6-8988*x**5+6657*x**4-2646*x**3+441*x**2),x)
 

Output:

(110*x**2 - 210*x + 105)/(22*x**4 - 64*x**3 + 63*x**2 - 21*x + (x**2 - 2*x 
 + 1)*exp(x))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\frac {5 \, {\left (22 \, x^{2} - 42 \, x + 21\right )}}{22 \, x^{4} - 64 \, x^{3} + 63 \, x^{2} + {\left (x^{2} - 2 \, x + 1\right )} e^{x} - 21 \, x} \] Input:

integrate(((-110*x^4+430*x^3-645*x^2+430*x-105)*exp(x)-4840*x^5+20900*x^4- 
36120*x^3+31080*x^2-13230*x+2205)/((x^4-4*x^3+6*x^2-4*x+1)*exp(x)^2+(44*x^ 
6-216*x^5+426*x^4-422*x^3+210*x^2-42*x)*exp(x)+484*x^8-2816*x^7+6868*x^6-8 
988*x^5+6657*x^4-2646*x^3+441*x^2),x, algorithm="maxima")
 

Output:

5*(22*x^2 - 42*x + 21)/(22*x^4 - 64*x^3 + 63*x^2 + (x^2 - 2*x + 1)*e^x - 2 
1*x)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\frac {5 \, {\left (22 \, x^{2} - 42 \, x + 21\right )}}{22 \, x^{4} - 64 \, x^{3} + x^{2} e^{x} + 63 \, x^{2} - 2 \, x e^{x} - 21 \, x + e^{x}} \] Input:

integrate(((-110*x^4+430*x^3-645*x^2+430*x-105)*exp(x)-4840*x^5+20900*x^4- 
36120*x^3+31080*x^2-13230*x+2205)/((x^4-4*x^3+6*x^2-4*x+1)*exp(x)^2+(44*x^ 
6-216*x^5+426*x^4-422*x^3+210*x^2-42*x)*exp(x)+484*x^8-2816*x^7+6868*x^6-8 
988*x^5+6657*x^4-2646*x^3+441*x^2),x, algorithm="giac")
 

Output:

5*(22*x^2 - 42*x + 21)/(22*x^4 - 64*x^3 + x^2*e^x + 63*x^2 - 2*x*e^x - 21* 
x + e^x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\int -\frac {13230\,x+{\mathrm {e}}^x\,\left (110\,x^4-430\,x^3+645\,x^2-430\,x+105\right )-31080\,x^2+36120\,x^3-20900\,x^4+4840\,x^5-2205}{{\mathrm {e}}^{2\,x}\,\left (x^4-4\,x^3+6\,x^2-4\,x+1\right )-{\mathrm {e}}^x\,\left (-44\,x^6+216\,x^5-426\,x^4+422\,x^3-210\,x^2+42\,x\right )+441\,x^2-2646\,x^3+6657\,x^4-8988\,x^5+6868\,x^6-2816\,x^7+484\,x^8} \,d x \] Input:

int(-(13230*x + exp(x)*(645*x^2 - 430*x - 430*x^3 + 110*x^4 + 105) - 31080 
*x^2 + 36120*x^3 - 20900*x^4 + 4840*x^5 - 2205)/(exp(2*x)*(6*x^2 - 4*x - 4 
*x^3 + x^4 + 1) - exp(x)*(42*x - 210*x^2 + 422*x^3 - 426*x^4 + 216*x^5 - 4 
4*x^6) + 441*x^2 - 2646*x^3 + 6657*x^4 - 8988*x^5 + 6868*x^6 - 2816*x^7 + 
484*x^8),x)
 

Output:

int(-(13230*x + exp(x)*(645*x^2 - 430*x - 430*x^3 + 110*x^4 + 105) - 31080 
*x^2 + 36120*x^3 - 20900*x^4 + 4840*x^5 - 2205)/(exp(2*x)*(6*x^2 - 4*x - 4 
*x^3 + x^4 + 1) - exp(x)*(42*x - 210*x^2 + 422*x^3 - 426*x^4 + 216*x^5 - 4 
4*x^6) + 441*x^2 - 2646*x^3 + 6657*x^4 - 8988*x^5 + 6868*x^6 - 2816*x^7 + 
484*x^8), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.63 \[ \int \frac {2205-13230 x+31080 x^2-36120 x^3+20900 x^4-4840 x^5+e^x \left (-105+430 x-645 x^2+430 x^3-110 x^4\right )}{441 x^2-2646 x^3+6657 x^4-8988 x^5+6868 x^6-2816 x^7+484 x^8+e^{2 x} \left (1-4 x+6 x^2-4 x^3+x^4\right )+e^x \left (-42 x+210 x^2-422 x^3+426 x^4-216 x^5+44 x^6\right )} \, dx=\frac {110 x^{2}-210 x +105}{e^{x} x^{2}-2 e^{x} x +e^{x}+22 x^{4}-64 x^{3}+63 x^{2}-21 x} \] Input:

int(((-110*x^4+430*x^3-645*x^2+430*x-105)*exp(x)-4840*x^5+20900*x^4-36120* 
x^3+31080*x^2-13230*x+2205)/((x^4-4*x^3+6*x^2-4*x+1)*exp(x)^2+(44*x^6-216* 
x^5+426*x^4-422*x^3+210*x^2-42*x)*exp(x)+484*x^8-2816*x^7+6868*x^6-8988*x^ 
5+6657*x^4-2646*x^3+441*x^2),x)
 

Output:

(5*(22*x**2 - 42*x + 21))/(e**x*x**2 - 2*e**x*x + e**x + 22*x**4 - 64*x**3 
 + 63*x**2 - 21*x)