\(\int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8)}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} (256 x^6+512 x^7+256 x^8)+e^x (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9)} \, dx\) [857]

Optimal result

Integrand size = 146, antiderivative size = 29 \[ \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=\frac {1+x}{-x \left (4 e^x+x\right )+\frac {3}{x^2 (4+4 x)}} \] Output:

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

Mathematica [A] (verified)

Time = 4.52 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21 \[ \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=-\frac {8 x^2 (1+x)^2}{-6+8 x^4+8 x^5+32 e^x x^3 (1+x)} \] Input:

Integrate[(24*x + 72*x^2 + 48*x^3 + 32*x^5 + 80*x^6 + 64*x^7 + 16*x^8 + E^ 
x*(64*x^4 + 192*x^5 + 256*x^6 + 192*x^7 + 64*x^8))/(9 - 24*x^4 - 24*x^5 + 
16*x^8 + 32*x^9 + 16*x^10 + E^(2*x)*(256*x^6 + 512*x^7 + 256*x^8) + E^x*(- 
96*x^3 - 96*x^4 + 128*x^7 + 256*x^8 + 128*x^9)),x]
 

Output:

(-8*x^2*(1 + x)^2)/(-6 + 8*x^4 + 8*x^5 + 32*E^x*x^3*(1 + x))
 

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.

\(\displaystyle \int \frac {16 x^8+64 x^7+80 x^6+32 x^5+48 x^3+72 x^2+e^x \left (64 x^8+192 x^7+256 x^6+192 x^5+64 x^4\right )+24 x}{16 x^{10}+32 x^9+16 x^8-24 x^5-24 x^4+e^{2 x} \left (256 x^8+512 x^7+256 x^6\right )+e^x \left (128 x^9+256 x^8+128 x^7-96 x^4-96 x^3\right )+9} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {8 x (x+1) \left (\left (8 e^x+2\right ) x^6+2 \left (8 e^x+3\right ) x^5+4 \left (4 e^x+1\right ) x^4+8 e^x x^3+6 x+3\right )}{\left (-4 x^5-4 x^4-16 e^x (x+1) x^3+3\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 8 \int \frac {x (x+1) \left (2 \left (1+4 e^x\right ) x^6+2 \left (3+8 e^x\right ) x^5+4 \left (1+4 e^x\right ) x^4+8 e^x x^3+6 x+3\right )}{\left (-4 x^5-4 x^4-16 e^x (x+1) x^3+3\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 8 \int \left (\frac {x \left (x^3+2 x^2+2 x+1\right )}{2 \left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )}-\frac {x \left (4 x^8+8 x^7-8 x^5-4 x^4-3 x^3-18 x^2-24 x-9\right )}{2 \left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 8 \left (\frac {9}{2} \int \frac {x}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+9 \int \frac {x^3}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+\frac {3}{2} \int \frac {x^4}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+2 \int \frac {x^5}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+\frac {1}{2} \int \frac {x}{4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3}dx+\int \frac {x^3}{4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3}dx+\frac {1}{2} \int \frac {x^4}{4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3}dx-2 \int \frac {x^9}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx-4 \int \frac {x^8}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+4 \int \frac {x^6}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+12 \int \frac {x^2}{\left (4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3\right )^2}dx+\int \frac {x^2}{4 x^5+16 e^x x^4+4 x^4+16 e^x x^3-3}dx\right )\)

Input:

Int[(24*x + 72*x^2 + 48*x^3 + 32*x^5 + 80*x^6 + 64*x^7 + 16*x^8 + E^x*(64* 
x^4 + 192*x^5 + 256*x^6 + 192*x^7 + 64*x^8))/(9 - 24*x^4 - 24*x^5 + 16*x^8 
 + 32*x^9 + 16*x^10 + E^(2*x)*(256*x^6 + 512*x^7 + 256*x^8) + E^x*(-96*x^3 
 - 96*x^4 + 128*x^7 + 256*x^8 + 128*x^9)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34

method	result	size
risch	\(-\frac {4 x^{2} \left (1+x \right )^{2}}{16 \,{\mathrm e}^{x} x^{4}+4 x^{5}+16 \,{\mathrm e}^{x} x^{3}+4 x^{4}-3}\)	\(39\)
norman	\(\frac {-4 x^{4}-8 x^{3}-4 x^{2}}{16 \,{\mathrm e}^{x} x^{4}+4 x^{5}+16 \,{\mathrm e}^{x} x^{3}+4 x^{4}-3}\)	\(46\)
parallelrisch	\(\frac {-64 x^{4}-128 x^{3}-64 x^{2}}{256 \,{\mathrm e}^{x} x^{4}+64 x^{5}+256 \,{\mathrm e}^{x} x^{3}+64 x^{4}-48}\)	\(47\)

Input:

int(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80*x^6+3 
2*x^5+48*x^3+72*x^2+24*x)/((256*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x^9+256 
*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24*x^4+9), 
x,method=_RETURNVERBOSE)
 

Output:

-4*x^2*(1+x)^2/(16*exp(x)*x^4+4*x^5+16*exp(x)*x^3+4*x^4-3)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=-\frac {4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}}{4 \, x^{5} + 4 \, x^{4} + 16 \, {\left (x^{4} + x^{3}\right )} e^{x} - 3} \] Input:

integrate(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80 
*x^6+32*x^5+48*x^3+72*x^2+24*x)/((256*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x 
^9+256*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24*x 
^4+9),x, algorithm="fricas")
 

Output:

-4*(x^4 + 2*x^3 + x^2)/(4*x^5 + 4*x^4 + 16*(x^4 + x^3)*e^x - 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 = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=\frac {- 4 x^{4} - 8 x^{3} - 4 x^{2}}{4 x^{5} + 4 x^{4} + \left (16 x^{4} + 16 x^{3}\right ) e^{x} - 3} \] Input:

integrate(((64*x**8+192*x**7+256*x**6+192*x**5+64*x**4)*exp(x)+16*x**8+64* 
x**7+80*x**6+32*x**5+48*x**3+72*x**2+24*x)/((256*x**8+512*x**7+256*x**6)*e 
xp(x)**2+(128*x**9+256*x**8+128*x**7-96*x**4-96*x**3)*exp(x)+16*x**10+32*x 
**9+16*x**8-24*x**5-24*x**4+9),x)
 

Output:

(-4*x**4 - 8*x**3 - 4*x**2)/(4*x**5 + 4*x**4 + (16*x**4 + 16*x**3)*exp(x) 
- 3)
 

Maxima [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=-\frac {4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}}{4 \, x^{5} + 4 \, x^{4} + 16 \, {\left (x^{4} + x^{3}\right )} e^{x} - 3} \] Input:

integrate(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80 
*x^6+32*x^5+48*x^3+72*x^2+24*x)/((256*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x 
^9+256*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24*x 
^4+9),x, algorithm="maxima")
 

Output:

-4*(x^4 + 2*x^3 + x^2)/(4*x^5 + 4*x^4 + 16*(x^4 + x^3)*e^x - 3)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=-\frac {4 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )}}{4 \, x^{5} + 16 \, x^{4} e^{x} + 4 \, x^{4} + 16 \, x^{3} e^{x} - 3} \] Input:

integrate(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80 
*x^6+32*x^5+48*x^3+72*x^2+24*x)/((256*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x 
^9+256*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24*x 
^4+9),x, algorithm="giac")
 

Output:

-4*(x^4 + 2*x^3 + x^2)/(4*x^5 + 16*x^4*e^x + 4*x^4 + 16*x^3*e^x - 3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=\int \frac {24\,x+{\mathrm {e}}^x\,\left (64\,x^8+192\,x^7+256\,x^6+192\,x^5+64\,x^4\right )+72\,x^2+48\,x^3+32\,x^5+80\,x^6+64\,x^7+16\,x^8}{{\mathrm {e}}^{2\,x}\,\left (256\,x^8+512\,x^7+256\,x^6\right )+{\mathrm {e}}^x\,\left (128\,x^9+256\,x^8+128\,x^7-96\,x^4-96\,x^3\right )-24\,x^4-24\,x^5+16\,x^8+32\,x^9+16\,x^{10}+9} \,d x \] Input:

int((24*x + exp(x)*(64*x^4 + 192*x^5 + 256*x^6 + 192*x^7 + 64*x^8) + 72*x^ 
2 + 48*x^3 + 32*x^5 + 80*x^6 + 64*x^7 + 16*x^8)/(exp(2*x)*(256*x^6 + 512*x 
^7 + 256*x^8) + exp(x)*(128*x^7 - 96*x^4 - 96*x^3 + 256*x^8 + 128*x^9) - 2 
4*x^4 - 24*x^5 + 16*x^8 + 32*x^9 + 16*x^10 + 9),x)
 

Output:

int((24*x + exp(x)*(64*x^4 + 192*x^5 + 256*x^6 + 192*x^7 + 64*x^8) + 72*x^ 
2 + 48*x^3 + 32*x^5 + 80*x^6 + 64*x^7 + 16*x^8)/(exp(2*x)*(256*x^6 + 512*x 
^7 + 256*x^8) + exp(x)*(128*x^7 - 96*x^4 - 96*x^3 + 256*x^8 + 128*x^9) - 2 
4*x^4 - 24*x^5 + 16*x^8 + 32*x^9 + 16*x^10 + 9), x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.55 \[ \int \frac {24 x+72 x^2+48 x^3+32 x^5+80 x^6+64 x^7+16 x^8+e^x \left (64 x^4+192 x^5+256 x^6+192 x^7+64 x^8\right )}{9-24 x^4-24 x^5+16 x^8+32 x^9+16 x^{10}+e^{2 x} \left (256 x^6+512 x^7+256 x^8\right )+e^x \left (-96 x^3-96 x^4+128 x^7+256 x^8+128 x^9\right )} \, dx=\frac {4 x^{2} \left (-x^{2}-2 x -1\right )}{16 e^{x} x^{4}+16 e^{x} x^{3}+4 x^{5}+4 x^{4}-3} \] Input:

int(((64*x^8+192*x^7+256*x^6+192*x^5+64*x^4)*exp(x)+16*x^8+64*x^7+80*x^6+3 
2*x^5+48*x^3+72*x^2+24*x)/((256*x^8+512*x^7+256*x^6)*exp(x)^2+(128*x^9+256 
*x^8+128*x^7-96*x^4-96*x^3)*exp(x)+16*x^10+32*x^9+16*x^8-24*x^5-24*x^4+9), 
x)
 

Output:

(4*x**2*( - x**2 - 2*x - 1))/(16*e**x*x**4 + 16*e**x*x**3 + 4*x**5 + 4*x** 
4 - 3)