3.10.47 \(\int \frac {-108-80 x+108 x^2+48 x^3-15 x^4-6 x^5+e^x (-108-108 x+108 x^2+36 x^3-15 x^4-3 x^5)}{11664 x^2+8640 x^3-6176 x^4-5472 x^5+984 x^6+1320 x^7+8 x^8-144 x^9-15 x^{10}+6 x^{11}+x^{12}+e^{2 x} (11664 x^2-7776 x^4+1944 x^6-216 x^8+9 x^{10})+e^x (23328 x^2+8640 x^3-15552 x^4-5472 x^5+3888 x^6+1320 x^7-432 x^8-144 x^9+18 x^{10}+6 x^{11})} \, dx\) [947]

3.10.47.1 Optimal result

Integrand size = 198, antiderivative size = 28 \[ \int \frac {-108-80 x+108 x^2+48 x^3-15 x^4-6 x^5+e^x \left (-108-108 x+108 x^2+36 x^3-15 x^4-3 x^5\right )}{11664 x^2+8640 x^3-6176 x^4-5472 x^5+984 x^6+1320 x^7+8 x^8-144 x^9-15 x^{10}+6 x^{11}+x^{12}+e^{2 x} \left (11664 x^2-7776 x^4+1944 x^6-216 x^8+9 x^{10}\right )+e^x \left (23328 x^2+8640 x^3-15552 x^4-5472 x^5+3888 x^6+1320 x^7-432 x^8-144 x^9+18 x^{10}+6 x^{11}\right )} \, dx=\frac {1}{x \left (4 x+\left (3+3 e^x+x\right ) \left (6-x^2\right )^2\right )} \]

output

1/(4*x+(-x^2+6)^2*(3+3*exp(x)+x))/x
 

3.10.47.2 Mathematica [A] (verified)

Time = 3.90 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.46 \[ \int \frac {-108-80 x+108 x^2+48 x^3-15 x^4-6 x^5+e^x \left (-108-108 x+108 x^2+36 x^3-15 x^4-3 x^5\right )}{11664 x^2+8640 x^3-6176 x^4-5472 x^5+984 x^6+1320 x^7+8 x^8-144 x^9-15 x^{10}+6 x^{11}+x^{12}+e^{2 x} \left (11664 x^2-7776 x^4+1944 x^6-216 x^8+9 x^{10}\right )+e^x \left (23328 x^2+8640 x^3-15552 x^4-5472 x^5+3888 x^6+1320 x^7-432 x^8-144 x^9+18 x^{10}+6 x^{11}\right )} \, dx=\frac {1}{x \left (108+40 x-36 x^2-12 x^3+3 x^4+x^5+3 e^x \left (-6+x^2\right )^2\right )} \]

input

Integrate[(-108 - 80*x + 108*x^2 + 48*x^3 - 15*x^4 - 6*x^5 + E^x*(-108 - 1 
08*x + 108*x^2 + 36*x^3 - 15*x^4 - 3*x^5))/(11664*x^2 + 8640*x^3 - 6176*x^ 
4 - 5472*x^5 + 984*x^6 + 1320*x^7 + 8*x^8 - 144*x^9 - 15*x^10 + 6*x^11 + x 
^12 + E^(2*x)*(11664*x^2 - 7776*x^4 + 1944*x^6 - 216*x^8 + 9*x^10) + E^x*( 
23328*x^2 + 8640*x^3 - 15552*x^4 - 5472*x^5 + 3888*x^6 + 1320*x^7 - 432*x^ 
8 - 144*x^9 + 18*x^10 + 6*x^11)),x]
 

output

1/(x*(108 + 40*x - 36*x^2 - 12*x^3 + 3*x^4 + x^5 + 3*E^x*(-6 + x^2)^2))
 

3.10.47.3 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.

\(\displaystyle \int \frac {-6 x^5-15 x^4+48 x^3+108 x^2+e^x \left (-3 x^5-15 x^4+36 x^3+108 x^2-108 x-108\right )-80 x-108}{x^{12}+6 x^{11}-15 x^{10}-144 x^9+8 x^8+1320 x^7+984 x^6-5472 x^5-6176 x^4+8640 x^3+11664 x^2+e^{2 x} \left (9 x^{10}-216 x^8+1944 x^6-7776 x^4+11664 x^2\right )+e^x \left (6 x^{11}+18 x^{10}-144 x^9-432 x^8+1320 x^7+3888 x^6-5472 x^5-15552 x^4+8640 x^3+23328 x^2\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-6 x^5-15 x^4+48 x^3+108 x^2-3 e^x \left (x^5+5 x^4-12 x^3-36 x^2+36 x+36\right )-80 x-108}{x^2 \left (x^5+3 x^4-12 x^3-36 x^2+3 e^x \left (x^2-6\right )^2+40 x+108\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x^7+2 x^6-18 x^5-36 x^4+112 x^3+228 x^2-240 x-408}{x \left (x^2-6\right ) \left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )^2}-\frac {x^3+5 x^2-6 x-6}{x^2 \left (x^2-6\right ) \left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 40 \int \frac {1}{\left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )^2}dx-8 \int \frac {1}{\left (\sqrt {6}-x\right ) \left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )^2}dx+68 \int \frac {1}{x \left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )^2}dx-24 \int \frac {x}{\left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )^2}dx-12 \int \frac {x^2}{\left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )^2}dx+2 \int \frac {x^3}{\left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )^2}dx+\int \frac {x^4}{\left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )^2}dx+8 \int \frac {1}{\left (x+\sqrt {6}\right ) \left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )^2}dx+\sqrt {\frac {2}{3}} \int \frac {1}{\left (\sqrt {6}-x\right ) \left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )}dx-\int \frac {1}{x^2 \left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )}dx-\int \frac {1}{x \left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )}dx+\sqrt {\frac {2}{3}} \int \frac {1}{\left (x+\sqrt {6}\right ) \left (x^5+3 e^x x^4+3 x^4-12 x^3-36 e^x x^2-36 x^2+40 x+108 e^x+108\right )}dx\)

input

Int[(-108 - 80*x + 108*x^2 + 48*x^3 - 15*x^4 - 6*x^5 + E^x*(-108 - 108*x + 
 108*x^2 + 36*x^3 - 15*x^4 - 3*x^5))/(11664*x^2 + 8640*x^3 - 6176*x^4 - 54 
72*x^5 + 984*x^6 + 1320*x^7 + 8*x^8 - 144*x^9 - 15*x^10 + 6*x^11 + x^12 + 
E^(2*x)*(11664*x^2 - 7776*x^4 + 1944*x^6 - 216*x^8 + 9*x^10) + E^x*(23328* 
x^2 + 8640*x^3 - 15552*x^4 - 5472*x^5 + 3888*x^6 + 1320*x^7 - 432*x^8 - 14 
4*x^9 + 18*x^10 + 6*x^11)),x]
 

output

$Aborted
 

3.10.47.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.10.47.4 Maple [A] (verified)

Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71

method	result	size
risch	\(\frac {1}{x \left (3 \,{\mathrm e}^{x} x^{4}+x^{5}+3 x^{4}-36 \,{\mathrm e}^{x} x^{2}-12 x^{3}-36 x^{2}+108 \,{\mathrm e}^{x}+40 x +108\right )}\)	\(48\)
parallelrisch	\(\frac {1}{x \left (3 \,{\mathrm e}^{x} x^{4}+x^{5}+3 x^{4}-36 \,{\mathrm e}^{x} x^{2}-12 x^{3}-36 x^{2}+108 \,{\mathrm e}^{x}+40 x +108\right )}\)	\(48\)

input

int(((-3*x^5-15*x^4+36*x^3+108*x^2-108*x-108)*exp(x)-6*x^5-15*x^4+48*x^3+1 
08*x^2-80*x-108)/((9*x^10-216*x^8+1944*x^6-7776*x^4+11664*x^2)*exp(x)^2+(6 
*x^11+18*x^10-144*x^9-432*x^8+1320*x^7+3888*x^6-5472*x^5-15552*x^4+8640*x^ 
3+23328*x^2)*exp(x)+x^12+6*x^11-15*x^10-144*x^9+8*x^8+1320*x^7+984*x^6-547 
2*x^5-6176*x^4+8640*x^3+11664*x^2),x,method=_RETURNVERBOSE)
 

output

1/x/(3*exp(x)*x^4+x^5+3*x^4-36*exp(x)*x^2-12*x^3-36*x^2+108*exp(x)+40*x+10 
8)
 

3.10.47.5 Fricas [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {-108-80 x+108 x^2+48 x^3-15 x^4-6 x^5+e^x \left (-108-108 x+108 x^2+36 x^3-15 x^4-3 x^5\right )}{11664 x^2+8640 x^3-6176 x^4-5472 x^5+984 x^6+1320 x^7+8 x^8-144 x^9-15 x^{10}+6 x^{11}+x^{12}+e^{2 x} \left (11664 x^2-7776 x^4+1944 x^6-216 x^8+9 x^{10}\right )+e^x \left (23328 x^2+8640 x^3-15552 x^4-5472 x^5+3888 x^6+1320 x^7-432 x^8-144 x^9+18 x^{10}+6 x^{11}\right )} \, dx=\frac {1}{x^{6} + 3 \, x^{5} - 12 \, x^{4} - 36 \, x^{3} + 40 \, x^{2} + 3 \, {\left (x^{5} - 12 \, x^{3} + 36 \, x\right )} e^{x} + 108 \, x} \]

input

integrate(((-3*x^5-15*x^4+36*x^3+108*x^2-108*x-108)*exp(x)-6*x^5-15*x^4+48 
*x^3+108*x^2-80*x-108)/((9*x^10-216*x^8+1944*x^6-7776*x^4+11664*x^2)*exp(x 
)^2+(6*x^11+18*x^10-144*x^9-432*x^8+1320*x^7+3888*x^6-5472*x^5-15552*x^4+8 
640*x^3+23328*x^2)*exp(x)+x^12+6*x^11-15*x^10-144*x^9+8*x^8+1320*x^7+984*x 
^6-5472*x^5-6176*x^4+8640*x^3+11664*x^2),x, algorithm=\
 

output

1/(x^6 + 3*x^5 - 12*x^4 - 36*x^3 + 40*x^2 + 3*(x^5 - 12*x^3 + 36*x)*e^x + 
108*x)
 

3.10.47.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (20) = 40\).

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.57 \[ \int \frac {-108-80 x+108 x^2+48 x^3-15 x^4-6 x^5+e^x \left (-108-108 x+108 x^2+36 x^3-15 x^4-3 x^5\right )}{11664 x^2+8640 x^3-6176 x^4-5472 x^5+984 x^6+1320 x^7+8 x^8-144 x^9-15 x^{10}+6 x^{11}+x^{12}+e^{2 x} \left (11664 x^2-7776 x^4+1944 x^6-216 x^8+9 x^{10}\right )+e^x \left (23328 x^2+8640 x^3-15552 x^4-5472 x^5+3888 x^6+1320 x^7-432 x^8-144 x^9+18 x^{10}+6 x^{11}\right )} \, dx=\frac {1}{x^{6} + 3 x^{5} - 12 x^{4} - 36 x^{3} + 40 x^{2} + 108 x + \left (3 x^{5} - 36 x^{3} + 108 x\right ) e^{x}} \]

input

integrate(((-3*x**5-15*x**4+36*x**3+108*x**2-108*x-108)*exp(x)-6*x**5-15*x 
**4+48*x**3+108*x**2-80*x-108)/((9*x**10-216*x**8+1944*x**6-7776*x**4+1166 
4*x**2)*exp(x)**2+(6*x**11+18*x**10-144*x**9-432*x**8+1320*x**7+3888*x**6- 
5472*x**5-15552*x**4+8640*x**3+23328*x**2)*exp(x)+x**12+6*x**11-15*x**10-1 
44*x**9+8*x**8+1320*x**7+984*x**6-5472*x**5-6176*x**4+8640*x**3+11664*x**2 
),x)
 

output

1/(x**6 + 3*x**5 - 12*x**4 - 36*x**3 + 40*x**2 + 108*x + (3*x**5 - 36*x**3 
 + 108*x)*exp(x))
 

3.10.47.7 Maxima [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {-108-80 x+108 x^2+48 x^3-15 x^4-6 x^5+e^x \left (-108-108 x+108 x^2+36 x^3-15 x^4-3 x^5\right )}{11664 x^2+8640 x^3-6176 x^4-5472 x^5+984 x^6+1320 x^7+8 x^8-144 x^9-15 x^{10}+6 x^{11}+x^{12}+e^{2 x} \left (11664 x^2-7776 x^4+1944 x^6-216 x^8+9 x^{10}\right )+e^x \left (23328 x^2+8640 x^3-15552 x^4-5472 x^5+3888 x^6+1320 x^7-432 x^8-144 x^9+18 x^{10}+6 x^{11}\right )} \, dx=\frac {1}{x^{6} + 3 \, x^{5} - 12 \, x^{4} - 36 \, x^{3} + 40 \, x^{2} + 3 \, {\left (x^{5} - 12 \, x^{3} + 36 \, x\right )} e^{x} + 108 \, x} \]

input

integrate(((-3*x^5-15*x^4+36*x^3+108*x^2-108*x-108)*exp(x)-6*x^5-15*x^4+48 
*x^3+108*x^2-80*x-108)/((9*x^10-216*x^8+1944*x^6-7776*x^4+11664*x^2)*exp(x 
)^2+(6*x^11+18*x^10-144*x^9-432*x^8+1320*x^7+3888*x^6-5472*x^5-15552*x^4+8 
640*x^3+23328*x^2)*exp(x)+x^12+6*x^11-15*x^10-144*x^9+8*x^8+1320*x^7+984*x 
^6-5472*x^5-6176*x^4+8640*x^3+11664*x^2),x, algorithm=\
 

output

1/(x^6 + 3*x^5 - 12*x^4 - 36*x^3 + 40*x^2 + 3*(x^5 - 12*x^3 + 36*x)*e^x + 
108*x)
 

3.10.47.8 Giac [A] (verification not implemented)

Time = 0.34 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {-108-80 x+108 x^2+48 x^3-15 x^4-6 x^5+e^x \left (-108-108 x+108 x^2+36 x^3-15 x^4-3 x^5\right )}{11664 x^2+8640 x^3-6176 x^4-5472 x^5+984 x^6+1320 x^7+8 x^8-144 x^9-15 x^{10}+6 x^{11}+x^{12}+e^{2 x} \left (11664 x^2-7776 x^4+1944 x^6-216 x^8+9 x^{10}\right )+e^x \left (23328 x^2+8640 x^3-15552 x^4-5472 x^5+3888 x^6+1320 x^7-432 x^8-144 x^9+18 x^{10}+6 x^{11}\right )} \, dx=\frac {1}{x^{6} + 3 \, x^{5} e^{x} + 3 \, x^{5} - 12 \, x^{4} - 36 \, x^{3} e^{x} - 36 \, x^{3} + 40 \, x^{2} + 108 \, x e^{x} + 108 \, x} \]

input

integrate(((-3*x^5-15*x^4+36*x^3+108*x^2-108*x-108)*exp(x)-6*x^5-15*x^4+48 
*x^3+108*x^2-80*x-108)/((9*x^10-216*x^8+1944*x^6-7776*x^4+11664*x^2)*exp(x 
)^2+(6*x^11+18*x^10-144*x^9-432*x^8+1320*x^7+3888*x^6-5472*x^5-15552*x^4+8 
640*x^3+23328*x^2)*exp(x)+x^12+6*x^11-15*x^10-144*x^9+8*x^8+1320*x^7+984*x 
^6-5472*x^5-6176*x^4+8640*x^3+11664*x^2),x, algorithm=\
 

output

1/(x^6 + 3*x^5*e^x + 3*x^5 - 12*x^4 - 36*x^3*e^x - 36*x^3 + 40*x^2 + 108*x 
*e^x + 108*x)
 

3.10.47.9 Mupad [F(-1)]

Timed out. \[ \int \frac {-108-80 x+108 x^2+48 x^3-15 x^4-6 x^5+e^x \left (-108-108 x+108 x^2+36 x^3-15 x^4-3 x^5\right )}{11664 x^2+8640 x^3-6176 x^4-5472 x^5+984 x^6+1320 x^7+8 x^8-144 x^9-15 x^{10}+6 x^{11}+x^{12}+e^{2 x} \left (11664 x^2-7776 x^4+1944 x^6-216 x^8+9 x^{10}\right )+e^x \left (23328 x^2+8640 x^3-15552 x^4-5472 x^5+3888 x^6+1320 x^7-432 x^8-144 x^9+18 x^{10}+6 x^{11}\right )} \, dx=\int -\frac {80\,x+{\mathrm {e}}^x\,\left (3\,x^5+15\,x^4-36\,x^3-108\,x^2+108\,x+108\right )-108\,x^2-48\,x^3+15\,x^4+6\,x^5+108}{{\mathrm {e}}^{2\,x}\,\left (9\,x^{10}-216\,x^8+1944\,x^6-7776\,x^4+11664\,x^2\right )+{\mathrm {e}}^x\,\left (6\,x^{11}+18\,x^{10}-144\,x^9-432\,x^8+1320\,x^7+3888\,x^6-5472\,x^5-15552\,x^4+8640\,x^3+23328\,x^2\right )+11664\,x^2+8640\,x^3-6176\,x^4-5472\,x^5+984\,x^6+1320\,x^7+8\,x^8-144\,x^9-15\,x^{10}+6\,x^{11}+x^{12}} \,d x \]

input

int(-(80*x + exp(x)*(108*x - 108*x^2 - 36*x^3 + 15*x^4 + 3*x^5 + 108) - 10 
8*x^2 - 48*x^3 + 15*x^4 + 6*x^5 + 108)/(exp(2*x)*(11664*x^2 - 7776*x^4 + 1 
944*x^6 - 216*x^8 + 9*x^10) + exp(x)*(23328*x^2 + 8640*x^3 - 15552*x^4 - 5 
472*x^5 + 3888*x^6 + 1320*x^7 - 432*x^8 - 144*x^9 + 18*x^10 + 6*x^11) + 11 
664*x^2 + 8640*x^3 - 6176*x^4 - 5472*x^5 + 984*x^6 + 1320*x^7 + 8*x^8 - 14 
4*x^9 - 15*x^10 + 6*x^11 + x^12),x)
 

output

int(-(80*x + exp(x)*(108*x - 108*x^2 - 36*x^3 + 15*x^4 + 3*x^5 + 108) - 10 
8*x^2 - 48*x^3 + 15*x^4 + 6*x^5 + 108)/(exp(2*x)*(11664*x^2 - 7776*x^4 + 1 
944*x^6 - 216*x^8 + 9*x^10) + exp(x)*(23328*x^2 + 8640*x^3 - 15552*x^4 - 5 
472*x^5 + 3888*x^6 + 1320*x^7 - 432*x^8 - 144*x^9 + 18*x^10 + 6*x^11) + 11 
664*x^2 + 8640*x^3 - 6176*x^4 - 5472*x^5 + 984*x^6 + 1320*x^7 + 8*x^8 - 14 
4*x^9 - 15*x^10 + 6*x^11 + x^12), x)