3.7.39 \(\int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6)}{784-280 x-31 x^2+10 x^3+x^4} \, dx\) [639]

3.7.39.1 Optimal result

Integrand size = 109, antiderivative size = 36 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{e^{x \left (5+\frac {x}{-x+\frac {5 (5-x)}{-\frac {3}{x}+x}}\right )} x^2} \]

output

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

3.7.39.2 Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{e^{5+4 x-\frac {10 (-14+5 x)}{-28+5 x+x^2}} x^2} \]

input

Integrate[(E^(E^((-137*x + 25*x^2 + 4*x^3)/(-28 + 5*x + x^2))*x^2 + (-137* 
x + 25*x^2 + 4*x^3)/(-28 + 5*x + x^2))*(1568*x + 3276*x^2 - 1462*x^3 - 54* 
x^4 + 42*x^5 + 4*x^6))/(784 - 280*x - 31*x^2 + 10*x^3 + x^4),x]
 

output

E^(E^(5 + 4*x - (10*(-14 + 5*x))/(-28 + 5*x + x^2))*x^2)
 

3.7.39.3 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.

\(\displaystyle \int \frac {\left (4 x^6+42 x^5-54 x^4-1462 x^3+3276 x^2+1568 x\right ) \exp \left (e^{\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}} x^2+\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}\right )}{x^4+10 x^3-31 x^2-280 x+784} \, dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {4 \left (4 x^6+42 x^5-54 x^4-1462 x^3+3276 x^2+1568 x\right ) \exp \left (e^{\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}} x^2+\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}\right )}{137 \sqrt {137} \left (-2 x+\sqrt {137}-5\right )}+\frac {4 \left (4 x^6+42 x^5-54 x^4-1462 x^3+3276 x^2+1568 x\right ) \exp \left (e^{\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}} x^2+\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}\right )}{137 \sqrt {137} \left (2 x+\sqrt {137}+5\right )}+\frac {4 \left (4 x^6+42 x^5-54 x^4-1462 x^3+3276 x^2+1568 x\right ) \exp \left (e^{\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}} x^2+\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}\right )}{137 \left (-2 x+\sqrt {137}-5\right )^2}+\frac {4 \left (4 x^6+42 x^5-54 x^4-1462 x^3+3276 x^2+1568 x\right ) \exp \left (e^{\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}} x^2+\frac {4 x^3+25 x^2-137 x}{x^2+5 x-28}\right )}{137 \left (2 x+\sqrt {137}+5\right )^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 x \left (2 x^5+21 x^4-27 x^3-731 x^2+1638 x+784\right ) \exp \left (-\frac {x \left (4 x^2+e^{\frac {x \left (4 x^2+25 x-137\right )}{x^2+5 x-28}} \left (x^2+5 x-28\right ) x+25 x-137\right )}{-x^2-5 x+28}\right )}{\left (-x^2-5 x+28\right )^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int \frac {\exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right ) x \left (2 x^5+21 x^4-27 x^3-731 x^2+1638 x+784\right )}{\left (-x^2-5 x+28\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 \int \left (2 \exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right ) x^2+\exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right ) x+25 \exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right )-\frac {15 \exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right ) (26 x-205)}{x^2+5 x-28}-\frac {5 \exp \left (\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}\right ) (3859 x-13300)}{\left (x^2+5 x-28\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (25 \int e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}dx+\frac {38590}{137} \left (5-\sqrt {137}\right ) \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{\left (-2 x+\sqrt {137}-5\right )^2}dx+\frac {266000}{137} \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{\left (-2 x+\sqrt {137}-5\right )^2}dx+\frac {3350 \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{-2 x+\sqrt {137}-5}dx}{\sqrt {137}}+\int e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}} xdx+2 \int e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}} x^2dx-\frac {30}{137} \left (1781-270 \sqrt {137}\right ) \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{2 x-\sqrt {137}+5}dx+\frac {38590}{137} \left (5+\sqrt {137}\right ) \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{\left (2 x+\sqrt {137}+5\right )^2}dx+\frac {266000}{137} \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{\left (2 x+\sqrt {137}+5\right )^2}dx-\frac {30}{137} \left (1781+270 \sqrt {137}\right ) \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{2 x+\sqrt {137}+5}dx+\frac {3350 \int \frac {e^{\frac {x \left (-4 x^2+e^{\frac {x \left (-4 x^2-25 x+137\right )}{-x^2-5 x+28}} \left (-x^2-5 x+28\right ) x-25 x+137\right )}{-x^2-5 x+28}}}{2 x+\sqrt {137}+5}dx}{\sqrt {137}}\right )\)

input

Int[(E^(E^((-137*x + 25*x^2 + 4*x^3)/(-28 + 5*x + x^2))*x^2 + (-137*x + 25 
*x^2 + 4*x^3)/(-28 + 5*x + x^2))*(1568*x + 3276*x^2 - 1462*x^3 - 54*x^4 + 
42*x^5 + 4*x^6))/(784 - 280*x - 31*x^2 + 10*x^3 + x^4),x]
 

output

$Aborted
 

3.7.39.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

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

rule 2463

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr 
and[u, Qx^p, x], x] /;  !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt 
Q[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x] && ILtQ[p, 
0]
 

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] 
]
 

3.7.39.4 Maple [A] (verified)

Time = 16.86 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81

method	result	size
risch	\({\mathrm e}^{x^{2} {\mathrm e}^{\frac {x \left (4 x^{2}+25 x -137\right )}{x^{2}+5 x -28}}}\)	\(29\)
norman	\(\frac {x^{2} {\mathrm e}^{x^{2} {\mathrm e}^{\frac {4 x^{3}+25 x^{2}-137 x}{x^{2}+5 x -28}}}+5 x \,{\mathrm e}^{x^{2} {\mathrm e}^{\frac {4 x^{3}+25 x^{2}-137 x}{x^{2}+5 x -28}}}-28 \,{\mathrm e}^{x^{2} {\mathrm e}^{\frac {4 x^{3}+25 x^{2}-137 x}{x^{2}+5 x -28}}}}{x^{2}+5 x -28}\)	\(115\)

input

int((4*x^6+42*x^5-54*x^4-1462*x^3+3276*x^2+1568*x)*exp((4*x^3+25*x^2-137*x 
)/(x^2+5*x-28))*exp(x^2*exp((4*x^3+25*x^2-137*x)/(x^2+5*x-28)))/(x^4+10*x^ 
3-31*x^2-280*x+784),x,method=_RETURNVERBOSE)
 

output

exp(x^2*exp(x*(4*x^2+25*x-137)/(x^2+5*x-28)))
 

3.7.39.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (31) = 62\).

Time = 0.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.61 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{\left (\frac {4 \, x^{3} + 25 \, x^{2} + {\left (x^{4} + 5 \, x^{3} - 28 \, x^{2}\right )} e^{\left (\frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )} - 137 \, x}{x^{2} + 5 \, x - 28} - \frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )} \]

input

integrate((4*x^6+42*x^5-54*x^4-1462*x^3+3276*x^2+1568*x)*exp((4*x^3+25*x^2 
-137*x)/(x^2+5*x-28))*exp(x^2*exp((4*x^3+25*x^2-137*x)/(x^2+5*x-28)))/(x^4 
+10*x^3-31*x^2-280*x+784),x, algorithm=\
 

output

e^((4*x^3 + 25*x^2 + (x^4 + 5*x^3 - 28*x^2)*e^((4*x^3 + 25*x^2 - 137*x)/(x 
^2 + 5*x - 28)) - 137*x)/(x^2 + 5*x - 28) - (4*x^3 + 25*x^2 - 137*x)/(x^2 
+ 5*x - 28))
 

3.7.39.6 Sympy [A] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{x^{2} e^{\frac {4 x^{3} + 25 x^{2} - 137 x}{x^{2} + 5 x - 28}}} \]

input

integrate((4*x**6+42*x**5-54*x**4-1462*x**3+3276*x**2+1568*x)*exp((4*x**3+ 
25*x**2-137*x)/(x**2+5*x-28))*exp(x**2*exp((4*x**3+25*x**2-137*x)/(x**2+5* 
x-28)))/(x**4+10*x**3-31*x**2-280*x+784),x)
 

output

exp(x**2*exp((4*x**3 + 25*x**2 - 137*x)/(x**2 + 5*x - 28)))
 

3.7.39.7 Maxima [A] (verification not implemented)

Time = 0.64 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=e^{\left (x^{2} e^{\left (4 \, x - \frac {50 \, x}{x^{2} + 5 \, x - 28} + \frac {140}{x^{2} + 5 \, x - 28} + 5\right )}\right )} \]

input

integrate((4*x^6+42*x^5-54*x^4-1462*x^3+3276*x^2+1568*x)*exp((4*x^3+25*x^2 
-137*x)/(x^2+5*x-28))*exp(x^2*exp((4*x^3+25*x^2-137*x)/(x^2+5*x-28)))/(x^4 
+10*x^3-31*x^2-280*x+784),x, algorithm=\
 

output

e^(x^2*e^(4*x - 50*x/(x^2 + 5*x - 28) + 140/(x^2 + 5*x - 28) + 5))
 

3.7.39.8 Giac [F]

\[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx=\int { \frac {2 \, {\left (2 \, x^{6} + 21 \, x^{5} - 27 \, x^{4} - 731 \, x^{3} + 1638 \, x^{2} + 784 \, x\right )} e^{\left (x^{2} e^{\left (\frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )} + \frac {4 \, x^{3} + 25 \, x^{2} - 137 \, x}{x^{2} + 5 \, x - 28}\right )}}{x^{4} + 10 \, x^{3} - 31 \, x^{2} - 280 \, x + 784} \,d x } \]

input

integrate((4*x^6+42*x^5-54*x^4-1462*x^3+3276*x^2+1568*x)*exp((4*x^3+25*x^2 
-137*x)/(x^2+5*x-28))*exp(x^2*exp((4*x^3+25*x^2-137*x)/(x^2+5*x-28)))/(x^4 
+10*x^3-31*x^2-280*x+784),x, algorithm=\
 

output

integrate(2*(2*x^6 + 21*x^5 - 27*x^4 - 731*x^3 + 1638*x^2 + 784*x)*e^(x^2* 
e^((4*x^3 + 25*x^2 - 137*x)/(x^2 + 5*x - 28)) + (4*x^3 + 25*x^2 - 137*x)/( 
x^2 + 5*x - 28))/(x^4 + 10*x^3 - 31*x^2 - 280*x + 784), x)
 

3.7.39.9 Mupad [B] (verification not implemented)

Time = 0.38 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {e^{e^{\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} x^2+\frac {-137 x+25 x^2+4 x^3}{-28+5 x+x^2}} \left (1568 x+3276 x^2-1462 x^3-54 x^4+42 x^5+4 x^6\right )}{784-280 x-31 x^2+10 x^3+x^4} \, dx={\mathrm {e}}^{x^2\,{\mathrm {e}}^{\frac {4\,x^3+25\,x^2-137\,x}{x^2+5\,x-28}}} \]

input

int((exp(x^2*exp((25*x^2 - 137*x + 4*x^3)/(5*x + x^2 - 28)))*exp((25*x^2 - 
 137*x + 4*x^3)/(5*x + x^2 - 28))*(1568*x + 3276*x^2 - 1462*x^3 - 54*x^4 + 
 42*x^5 + 4*x^6))/(10*x^3 - 31*x^2 - 280*x + x^4 + 784),x)
 

output

exp(x^2*exp((25*x^2 - 137*x + 4*x^3)/(5*x + x^2 - 28)))