\(\int \frac {23-2 x+529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} (18+18 x+828 x^2-36 x^3)+e^{16+2 x} (3+6 x+462 x^2-6 x^3)}{529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} (828 x^2-36 x^3)+e^{16+2 x} (462 x^2-6 x^3)} \, dx\) [2760]

Optimal result

Integrand size = 163, antiderivative size = 22 \[ \int \frac {23-2 x+529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (18+18 x+828 x^2-36 x^3\right )+e^{16+2 x} \left (3+6 x+462 x^2-6 x^3\right )}{529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (828 x^2-36 x^3\right )+e^{16+2 x} \left (462 x^2-6 x^3\right )} \, dx=x+\frac {1}{x \left (4-3 \left (3+e^{8+x}\right )^2+x\right )} \] Output:

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

Mathematica [A] (verified)

Time = 3.69 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {23-2 x+529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (18+18 x+828 x^2-36 x^3\right )+e^{16+2 x} \left (3+6 x+462 x^2-6 x^3\right )}{529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (828 x^2-36 x^3\right )+e^{16+2 x} \left (462 x^2-6 x^3\right )} \, dx=-\frac {1}{\left (23+18 e^{8+x}+3 e^{16+2 x}-x\right ) x}+x \] Input:

Integrate[(23 - 2*x + 529*x^2 + 108*E^(24 + 3*x)*x^2 + 9*E^(32 + 4*x)*x^2 
- 46*x^3 + x^4 + E^(8 + x)*(18 + 18*x + 828*x^2 - 36*x^3) + E^(16 + 2*x)*( 
3 + 6*x + 462*x^2 - 6*x^3))/(529*x^2 + 108*E^(24 + 3*x)*x^2 + 9*E^(32 + 4* 
x)*x^2 - 46*x^3 + x^4 + E^(8 + x)*(828*x^2 - 36*x^3) + E^(16 + 2*x)*(462*x 
^2 - 6*x^3)),x]
 

Output:

-(1/((23 + 18*E^(8 + x) + 3*E^(16 + 2*x) - x)*x)) + 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.

\(\displaystyle \int \frac {x^4-46 x^3+108 e^{3 x+24} x^2+9 e^{4 x+32} x^2+529 x^2+e^{x+8} \left (-36 x^3+828 x^2+18 x+18\right )+e^{2 x+16} \left (-6 x^3+462 x^2+6 x+3\right )-2 x+23}{x^4-46 x^3+108 e^{3 x+24} x^2+9 e^{4 x+32} x^2+529 x^2+e^{x+8} \left (828 x^2-36 x^3\right )+e^{2 x+16} \left (462 x^2-6 x^3\right )} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {x^4-46 x^3+108 e^{3 x+24} x^2+9 e^{4 x+32} x^2+529 x^2+e^{x+8} \left (-36 x^3+828 x^2+18 x+18\right )+e^{2 x+16} \left (-6 x^3+462 x^2+6 x+3\right )-2 x+23}{\left (-x+18 e^{x+8}+3 e^{2 x+16}+23\right )^2 x^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x+1}{\left (-x+18 e^{x+8}+3 e^{2 x+16}+23\right ) x^2}-\frac {-2 x+18 e^{x+8}+47}{\left (-x+18 e^{x+8}+3 e^{2 x+16}+23\right )^2 x}+1\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \int \frac {1}{\left (-x+18 e^{x+8}+3 e^{2 x+16}+23\right ) x^2}dx+2 \int \frac {1}{\left (-x+18 e^{x+8}+3 e^{2 x+16}+23\right )^2}dx-47 \int \frac {1}{\left (-x+18 e^{x+8}+3 e^{2 x+16}+23\right )^2 x}dx-18 \int \frac {e^{x+8}}{\left (-x+18 e^{x+8}+3 e^{2 x+16}+23\right )^2 x}dx+2 \int \frac {1}{\left (-x+18 e^{x+8}+3 e^{2 x+16}+23\right ) x}dx+x\)

Input:

Int[(23 - 2*x + 529*x^2 + 108*E^(24 + 3*x)*x^2 + 9*E^(32 + 4*x)*x^2 - 46*x 
^3 + x^4 + E^(8 + x)*(18 + 18*x + 828*x^2 - 36*x^3) + E^(16 + 2*x)*(3 + 6* 
x + 462*x^2 - 6*x^3))/(529*x^2 + 108*E^(24 + 3*x)*x^2 + 9*E^(32 + 4*x)*x^2 
 - 46*x^3 + x^4 + E^(8 + x)*(828*x^2 - 36*x^3) + E^(16 + 2*x)*(462*x^2 - 6 
*x^3)),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.98 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32

method	result	size
risch	\(x -\frac {1}{x \left (3 \,{\mathrm e}^{2 x +16}+18 \,{\mathrm e}^{x +8}-x +23\right )}\)	\(29\)
norman	\(\frac {-1+529 x +414 \,{\mathrm e}^{x} {\mathrm e}^{8} x +69 \,{\mathrm e}^{16} {\mathrm e}^{2 x} x -x^{3}+18 \,{\mathrm e}^{x} {\mathrm e}^{8} x^{2}+3 \,{\mathrm e}^{16} {\mathrm e}^{2 x} x^{2}}{x \left (3 \,{\mathrm e}^{2 x} {\mathrm e}^{16}+18 \,{\mathrm e}^{x} {\mathrm e}^{8}-x +23\right )}\)	\(84\)
parallelrisch	\(\frac {-1+529 x +414 \,{\mathrm e}^{x} {\mathrm e}^{8} x +69 \,{\mathrm e}^{16} {\mathrm e}^{2 x} x -x^{3}+18 \,{\mathrm e}^{x} {\mathrm e}^{8} x^{2}+3 \,{\mathrm e}^{16} {\mathrm e}^{2 x} x^{2}}{x \left (3 \,{\mathrm e}^{2 x} {\mathrm e}^{16}+18 \,{\mathrm e}^{x} {\mathrm e}^{8}-x +23\right )}\)	\(84\)

Input:

int((9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*x^3+462*x^2+6*x 
+3)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2+18*x+18)*exp(4)^2*exp(x)+x^4-46*x^3 
+529*x^2-2*x+23)/(9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*x^ 
3+462*x^2)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2)*exp(4)^2*exp(x)+x^4-46*x^3+ 
529*x^2),x,method=_RETURNVERBOSE)
 

Output:

x-1/x/(3*exp(2*x+16)+18*exp(x+8)-x+23)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).

Time = 0.07 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {23-2 x+529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (18+18 x+828 x^2-36 x^3\right )+e^{16+2 x} \left (3+6 x+462 x^2-6 x^3\right )}{529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (828 x^2-36 x^3\right )+e^{16+2 x} \left (462 x^2-6 x^3\right )} \, dx=\frac {x^{3} - 3 \, x^{2} e^{\left (2 \, x + 16\right )} - 18 \, x^{2} e^{\left (x + 8\right )} - 23 \, x^{2} + 1}{x^{2} - 3 \, x e^{\left (2 \, x + 16\right )} - 18 \, x e^{\left (x + 8\right )} - 23 \, x} \] Input:

integrate((9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*x^3+462*x 
^2+6*x+3)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2+18*x+18)*exp(4)^2*exp(x)+x^4- 
46*x^3+529*x^2-2*x+23)/(9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+ 
(-6*x^3+462*x^2)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2)*exp(4)^2*exp(x)+x^4-4 
6*x^3+529*x^2),x, algorithm="fricas")
 

Output:

(x^3 - 3*x^2*e^(2*x + 16) - 18*x^2*e^(x + 8) - 23*x^2 + 1)/(x^2 - 3*x*e^(2 
*x + 16) - 18*x*e^(x + 8) - 23*x)
 

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \frac {23-2 x+529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (18+18 x+828 x^2-36 x^3\right )+e^{16+2 x} \left (3+6 x+462 x^2-6 x^3\right )}{529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (828 x^2-36 x^3\right )+e^{16+2 x} \left (462 x^2-6 x^3\right )} \, dx=x - \frac {1}{- x^{2} + 3 x e^{16} e^{2 x} + 18 x e^{8} e^{x} + 23 x} \] Input:

integrate((9*x**2*exp(4)**8*exp(x)**4+108*x**2*exp(4)**6*exp(x)**3+(-6*x** 
3+462*x**2+6*x+3)*exp(4)**4*exp(x)**2+(-36*x**3+828*x**2+18*x+18)*exp(4)** 
2*exp(x)+x**4-46*x**3+529*x**2-2*x+23)/(9*x**2*exp(4)**8*exp(x)**4+108*x** 
2*exp(4)**6*exp(x)**3+(-6*x**3+462*x**2)*exp(4)**4*exp(x)**2+(-36*x**3+828 
*x**2)*exp(4)**2*exp(x)+x**4-46*x**3+529*x**2),x)
 

Output:

x - 1/(-x**2 + 3*x*exp(16)*exp(2*x) + 18*x*exp(8)*exp(x) + 23*x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).

Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {23-2 x+529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (18+18 x+828 x^2-36 x^3\right )+e^{16+2 x} \left (3+6 x+462 x^2-6 x^3\right )}{529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (828 x^2-36 x^3\right )+e^{16+2 x} \left (462 x^2-6 x^3\right )} \, dx=\frac {x^{3} - 3 \, x^{2} e^{\left (2 \, x + 16\right )} - 18 \, x^{2} e^{\left (x + 8\right )} - 23 \, x^{2} + 1}{x^{2} - 3 \, x e^{\left (2 \, x + 16\right )} - 18 \, x e^{\left (x + 8\right )} - 23 \, x} \] Input:

integrate((9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*x^3+462*x 
^2+6*x+3)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2+18*x+18)*exp(4)^2*exp(x)+x^4- 
46*x^3+529*x^2-2*x+23)/(9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+ 
(-6*x^3+462*x^2)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2)*exp(4)^2*exp(x)+x^4-4 
6*x^3+529*x^2),x, algorithm="maxima")
 

Output:

(x^3 - 3*x^2*e^(2*x + 16) - 18*x^2*e^(x + 8) - 23*x^2 + 1)/(x^2 - 3*x*e^(2 
*x + 16) - 18*x*e^(x + 8) - 23*x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).

Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \frac {23-2 x+529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (18+18 x+828 x^2-36 x^3\right )+e^{16+2 x} \left (3+6 x+462 x^2-6 x^3\right )}{529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (828 x^2-36 x^3\right )+e^{16+2 x} \left (462 x^2-6 x^3\right )} \, dx=\frac {x^{3} - 3 \, x^{2} e^{\left (2 \, x + 16\right )} - 18 \, x^{2} e^{\left (x + 8\right )} - 23 \, x^{2} + 2}{x^{2} - 3 \, x e^{\left (2 \, x + 16\right )} - 18 \, x e^{\left (x + 8\right )} - 23 \, x} \] Input:

integrate((9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*x^3+462*x 
^2+6*x+3)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2+18*x+18)*exp(4)^2*exp(x)+x^4- 
46*x^3+529*x^2-2*x+23)/(9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+ 
(-6*x^3+462*x^2)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2)*exp(4)^2*exp(x)+x^4-4 
6*x^3+529*x^2),x, algorithm="giac")
 

Output:

(x^3 - 3*x^2*e^(2*x + 16) - 18*x^2*e^(x + 8) - 23*x^2 + 2)/(x^2 - 3*x*e^(2 
*x + 16) - 18*x*e^(x + 8) - 23*x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {23-2 x+529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (18+18 x+828 x^2-36 x^3\right )+e^{16+2 x} \left (3+6 x+462 x^2-6 x^3\right )}{529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (828 x^2-36 x^3\right )+e^{16+2 x} \left (462 x^2-6 x^3\right )} \, dx=\int \frac {{\mathrm {e}}^{x+8}\,\left (-36\,x^3+828\,x^2+18\,x+18\right )-2\,x+{\mathrm {e}}^{2\,x+16}\,\left (-6\,x^3+462\,x^2+6\,x+3\right )+108\,x^2\,{\mathrm {e}}^{3\,x+24}+9\,x^2\,{\mathrm {e}}^{4\,x+32}+529\,x^2-46\,x^3+x^4+23}{{\mathrm {e}}^{x+8}\,\left (828\,x^2-36\,x^3\right )+{\mathrm {e}}^{2\,x+16}\,\left (462\,x^2-6\,x^3\right )+108\,x^2\,{\mathrm {e}}^{3\,x+24}+9\,x^2\,{\mathrm {e}}^{4\,x+32}+529\,x^2-46\,x^3+x^4} \,d x \] Input:

int((529*x^2 - 2*x - 46*x^3 + x^4 + exp(8)*exp(x)*(18*x + 828*x^2 - 36*x^3 
 + 18) + exp(2*x)*exp(16)*(6*x + 462*x^2 - 6*x^3 + 3) + 108*x^2*exp(3*x)*e 
xp(24) + 9*x^2*exp(4*x)*exp(32) + 23)/(529*x^2 - 46*x^3 + x^4 + exp(8)*exp 
(x)*(828*x^2 - 36*x^3) + exp(2*x)*exp(16)*(462*x^2 - 6*x^3) + 108*x^2*exp( 
3*x)*exp(24) + 9*x^2*exp(4*x)*exp(32)),x)
 

Output:

int((exp(x + 8)*(18*x + 828*x^2 - 36*x^3 + 18) - 2*x + exp(2*x + 16)*(6*x 
+ 462*x^2 - 6*x^3 + 3) + 108*x^2*exp(3*x + 24) + 9*x^2*exp(4*x + 32) + 529 
*x^2 - 46*x^3 + x^4 + 23)/(exp(x + 8)*(828*x^2 - 36*x^3) + exp(2*x + 16)*( 
462*x^2 - 6*x^3) + 108*x^2*exp(3*x + 24) + 9*x^2*exp(4*x + 32) + 529*x^2 - 
 46*x^3 + x^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.95 \[ \int \frac {23-2 x+529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (18+18 x+828 x^2-36 x^3\right )+e^{16+2 x} \left (3+6 x+462 x^2-6 x^3\right )}{529 x^2+108 e^{24+3 x} x^2+9 e^{32+4 x} x^2-46 x^3+x^4+e^{8+x} \left (828 x^2-36 x^3\right )+e^{16+2 x} \left (462 x^2-6 x^3\right )} \, dx=\frac {3 e^{2 x} e^{16} x^{2}+18 e^{x} e^{8} x^{2}-x^{3}+23 x^{2}-1}{x \left (3 e^{2 x} e^{16}+18 e^{x} e^{8}-x +23\right )} \] Input:

int((9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*x^3+462*x^2+6*x 
+3)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2+18*x+18)*exp(4)^2*exp(x)+x^4-46*x^3 
+529*x^2-2*x+23)/(9*x^2*exp(4)^8*exp(x)^4+108*x^2*exp(4)^6*exp(x)^3+(-6*x^ 
3+462*x^2)*exp(4)^4*exp(x)^2+(-36*x^3+828*x^2)*exp(4)^2*exp(x)+x^4-46*x^3+ 
529*x^2),x)
 

Output:

(3*e**(2*x)*e**16*x**2 + 18*e**x*e**8*x**2 - x**3 + 23*x**2 - 1)/(x*(3*e** 
(2*x)*e**16 + 18*e**x*e**8 - x + 23))