3.14.7 \(\int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2)}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} (162 x^3+180 x^4+50 x^5))}{81 x^3+90 x^4+25 x^5} \, dx\) [1307]

3.14.7.1 Optimal result

Integrand size = 174, antiderivative size = 28 \begin {dmath*} \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=e^{\frac {e^{x-\frac {5}{x+\frac {5}{4} x (1+x)}}}{x}+2 x} \end {dmath*}

output

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

3.14.7.2 Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14 \begin {dmath*} \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=e^{\frac {e^{-\frac {20}{9 x}+x+\frac {100}{9 (9+5 x)}}}{x}+2 x} \end {dmath*}

input

Integrate[(E^((1 + 2*E^((20 - 9*x^2 - 5*x^3)/(9*x + 5*x^2))*x^2)/(E^((20 - 
 9*x^2 - 5*x^3)/(9*x + 5*x^2))*x) - (20 - 9*x^2 - 5*x^3)/(9*x + 5*x^2))*(1 
80 + 119*x - 9*x^2 + 65*x^3 + 25*x^4 + E^((20 - 9*x^2 - 5*x^3)/(9*x + 5*x^ 
2))*(162*x^3 + 180*x^4 + 50*x^5)))/(81*x^3 + 90*x^4 + 25*x^5),x]
 

output

E^(E^(-20/(9*x) + x + 100/(9*(9 + 5*x)))/x + 2*x)
 

3.14.7.3 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.

\(\displaystyle \int \frac {\left (25 x^4+65 x^3-9 x^2+e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (50 x^5+180 x^4+162 x^3\right )+119 x+180\right ) \exp \left (\frac {e^{-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (2 e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} x^2+1\right )}{x}-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}\right )}{25 x^5+90 x^4+81 x^3} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (25 x^4+65 x^3-9 x^2+e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (50 x^5+180 x^4+162 x^3\right )+119 x+180\right ) \exp \left (\frac {e^{-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (2 e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} x^2+1\right )}{x}-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}\right )}{x^3 \left (25 x^2+90 x+81\right )}dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {\left (25 x^4+65 x^3-9 x^2+e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (50 x^5+180 x^4+162 x^3\right )+119 x+180\right ) \exp \left (\frac {e^{-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (2 e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} x^2+1\right )}{x}-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}\right )}{x^3 (5 x+9)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {25 x \exp \left (\frac {e^{-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (2 e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} x^2+1\right )}{x}-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}\right )}{(5 x+9)^2}+2 \exp \left (\frac {e^{-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (2 e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} x^2+1\right )}{x}+\frac {-5 x^3-9 x^2+20}{x (5 x+9)}-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}\right )+\frac {65 \exp \left (\frac {e^{-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (2 e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} x^2+1\right )}{x}-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}\right )}{(5 x+9)^2}-\frac {9 \exp \left (\frac {e^{-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (2 e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} x^2+1\right )}{x}-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}\right )}{(5 x+9)^2 x}+\frac {119 \exp \left (\frac {e^{-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (2 e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} x^2+1\right )}{x}-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}\right )}{(5 x+9)^2 x^2}+\frac {180 \exp \left (\frac {e^{-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} \left (2 e^{\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}} x^2+1\right )}{x}-\frac {-5 x^3-9 x^2+20}{5 x^2+9 x}\right )}{(5 x+9)^2 x^3}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {\left (2 e^{\frac {20}{5 x^2+9 x}} (5 x+9)^2 x^3+e^x \left (25 x^4+65 x^3-9 x^2+119 x+180\right )\right ) \exp \left (\frac {e^{x-\frac {20}{5 x^2+9 x}}}{x}+\frac {2 \left (5 x^3+9 x^2-10\right )}{x (5 x+9)}\right )}{x^3 (5 x+9)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (2 \exp \left (\frac {e^{x-\frac {20}{5 x^2+9 x}}}{x}+\frac {2 \left (5 x^3+9 x^2-10\right )}{x (5 x+9)}+\frac {20}{x (5 x+9)}\right )+\frac {\left (25 x^4+65 x^3-9 x^2+119 x+180\right ) \exp \left (\frac {e^{x-\frac {20}{5 x^2+9 x}}}{x}+\frac {2 \left (5 x^3+9 x^2-10\right )}{(5 x+9) x}+x\right )}{x^3 (5 x+9)^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {20}{9} \int \frac {\exp \left (x+\frac {e^{x-\frac {20}{5 x^2+9 x}}}{x}+\frac {2 \left (5 x^3+9 x^2-10\right )}{(5 x+9) x}\right )}{x^3}dx-\int \frac {\exp \left (x+\frac {e^{x-\frac {20}{5 x^2+9 x}}}{x}+\frac {2 \left (5 x^3+9 x^2-10\right )}{(5 x+9) x}\right )}{x^2}dx+\frac {229}{729} \int \frac {\exp \left (x+\frac {e^{x-\frac {20}{5 x^2+9 x}}}{x}+\frac {2 \left (5 x^3+9 x^2-10\right )}{(5 x+9) x}\right )}{x}dx+\frac {2500}{81} \int \frac {\exp \left (x+\frac {e^{x-\frac {20}{5 x^2+9 x}}}{x}+\frac {2 \left (5 x^3+9 x^2-10\right )}{(5 x+9) x}\right )}{(5 x+9)^2}dx+\frac {2500}{729} \int \frac {\exp \left (x+\frac {e^{x-\frac {20}{5 x^2+9 x}}}{x}+\frac {2 \left (5 x^3+9 x^2-10\right )}{(5 x+9) x}\right )}{5 x+9}dx+2 \int e^{2 x+\frac {e^{x-\frac {20}{5 x^2+9 x}}}{x}}dx\)

input

Int[(E^((1 + 2*E^((20 - 9*x^2 - 5*x^3)/(9*x + 5*x^2))*x^2)/(E^((20 - 9*x^2 
 - 5*x^3)/(9*x + 5*x^2))*x) - (20 - 9*x^2 - 5*x^3)/(9*x + 5*x^2))*(180 + 1 
19*x - 9*x^2 + 65*x^3 + 25*x^4 + E^((20 - 9*x^2 - 5*x^3)/(9*x + 5*x^2))*(1 
62*x^3 + 180*x^4 + 50*x^5)))/(81*x^3 + 90*x^4 + 25*x^5),x]
 

output

$Aborted
 

3.14.7.3.1 Defintions of rubi rules used

rule 2007

Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, 
x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex 
pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol 
yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
 

rule 2009

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

rule 2026

Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p 
*r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ 
erQ[p] &&  !MonomialQ[Px, x] && (ILtQ[p, 0] ||  !PolyQ[u, x])
 

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] 
]
 

3.14.7.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(61\) vs. \(2(28)=56\).

Time = 1.41 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21

method	result	size
risch	\({\mathrm e}^{\frac {\left (2 x^{2} {\mathrm e}^{-\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}+1\right ) {\mathrm e}^{\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}}{x}}\)	\(62\)
parallelrisch	\(\frac {22500 x^{2} {\mathrm e}^{\frac {\left (2 x^{2} {\mathrm e}^{-\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}+1\right ) {\mathrm e}^{\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}}{x}}+40500 x \,{\mathrm e}^{\frac {\left (2 x^{2} {\mathrm e}^{-\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}+1\right ) {\mathrm e}^{\frac {5 x^{3}+9 x^{2}-20}{x \left (5 x +9\right )}}}{x}}}{4500 x \left (5 x +9\right )}\)	\(150\)
norman	\(\frac {\left (9 x^{2} {\mathrm e}^{\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}} {\mathrm e}^{\frac {\left (2 x^{2} {\mathrm e}^{\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}}+1\right ) {\mathrm e}^{-\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}}}{x}}+5 x^{3} {\mathrm e}^{\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}} {\mathrm e}^{\frac {\left (2 x^{2} {\mathrm e}^{\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}}+1\right ) {\mathrm e}^{-\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}}}{x}}\right ) {\mathrm e}^{-\frac {-5 x^{3}-9 x^{2}+20}{5 x^{2}+9 x}}}{x^{2} \left (5 x +9\right )}\)	\(228\)

input

int(((50*x^5+180*x^4+162*x^3)*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+25*x^4+65 
*x^3-9*x^2+119*x+180)*exp((2*x^2*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+1)/x/e 
xp((-5*x^3-9*x^2+20)/(5*x^2+9*x)))/(25*x^5+90*x^4+81*x^3)/exp((-5*x^3-9*x^ 
2+20)/(5*x^2+9*x)),x,method=_RETURNVERBOSE)
 

output

exp((2*x^2*exp(-(5*x^3+9*x^2-20)/x/(5*x+9))+1)/x*exp((5*x^3+9*x^2-20)/x/(5 
*x+9)))
 

3.14.7.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (26) = 52\).

Time = 0.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.93 \begin {dmath*} \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=e^{\left (\frac {15 \, x^{3} + 27 \, x^{2} + {\left (5 \, x + 9\right )} e^{\left (\frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )} - 20}{5 \, x^{2} + 9 \, x} - \frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )} \end {dmath*}

input

integrate(((50*x^5+180*x^4+162*x^3)*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+25* 
x^4+65*x^3-9*x^2+119*x+180)*exp((2*x^2*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+ 
1)/x/exp((-5*x^3-9*x^2+20)/(5*x^2+9*x)))/(25*x^5+90*x^4+81*x^3)/exp((-5*x^ 
3-9*x^2+20)/(5*x^2+9*x)),x, algorithm=\
 

output

e^((15*x^3 + 27*x^2 + (5*x + 9)*e^((5*x^3 + 9*x^2 - 20)/(5*x^2 + 9*x)) - 2 
0)/(5*x^2 + 9*x) - (5*x^3 + 9*x^2 - 20)/(5*x^2 + 9*x))
 

3.14.7.6 Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 0.43 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \begin {dmath*} \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=e^{\frac {\left (2 x^{2} e^{\frac {- 5 x^{3} - 9 x^{2} + 20}{5 x^{2} + 9 x}} + 1\right ) e^{- \frac {- 5 x^{3} - 9 x^{2} + 20}{5 x^{2} + 9 x}}}{x}} \end {dmath*}

input

integrate(((50*x**5+180*x**4+162*x**3)*exp((-5*x**3-9*x**2+20)/(5*x**2+9*x 
))+25*x**4+65*x**3-9*x**2+119*x+180)*exp((2*x**2*exp((-5*x**3-9*x**2+20)/( 
5*x**2+9*x))+1)/x/exp((-5*x**3-9*x**2+20)/(5*x**2+9*x)))/(25*x**5+90*x**4+ 
81*x**3)/exp((-5*x**3-9*x**2+20)/(5*x**2+9*x)),x)
 

output

exp((2*x**2*exp((-5*x**3 - 9*x**2 + 20)/(5*x**2 + 9*x)) + 1)*exp(-(-5*x**3 
 - 9*x**2 + 20)/(5*x**2 + 9*x))/x)
 

3.14.7.7 Maxima [A] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \begin {dmath*} \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=e^{\left (2 \, x + \frac {e^{\left (x + \frac {100}{9 \, {\left (5 \, x + 9\right )}} - \frac {20}{9 \, x}\right )}}{x}\right )} \end {dmath*}

input

integrate(((50*x^5+180*x^4+162*x^3)*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+25* 
x^4+65*x^3-9*x^2+119*x+180)*exp((2*x^2*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+ 
1)/x/exp((-5*x^3-9*x^2+20)/(5*x^2+9*x)))/(25*x^5+90*x^4+81*x^3)/exp((-5*x^ 
3-9*x^2+20)/(5*x^2+9*x)),x, algorithm=\
 

output

e^(2*x + e^(x + 100/9/(5*x + 9) - 20/9/x)/x)
 

3.14.7.8 Giac [F]

\begin {dmath*} \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx=\int { \frac {{\left (25 \, x^{4} + 65 \, x^{3} - 9 \, x^{2} + 2 \, {\left (25 \, x^{5} + 90 \, x^{4} + 81 \, x^{3}\right )} e^{\left (-\frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )} + 119 \, x + 180\right )} e^{\left (\frac {{\left (2 \, x^{2} e^{\left (-\frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )} + 1\right )} e^{\left (\frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )}}{x} + \frac {5 \, x^{3} + 9 \, x^{2} - 20}{5 \, x^{2} + 9 \, x}\right )}}{25 \, x^{5} + 90 \, x^{4} + 81 \, x^{3}} \,d x } \end {dmath*}

input

integrate(((50*x^5+180*x^4+162*x^3)*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+25* 
x^4+65*x^3-9*x^2+119*x+180)*exp((2*x^2*exp((-5*x^3-9*x^2+20)/(5*x^2+9*x))+ 
1)/x/exp((-5*x^3-9*x^2+20)/(5*x^2+9*x)))/(25*x^5+90*x^4+81*x^3)/exp((-5*x^ 
3-9*x^2+20)/(5*x^2+9*x)),x, algorithm=\
 

output

integrate((25*x^4 + 65*x^3 - 9*x^2 + 2*(25*x^5 + 90*x^4 + 81*x^3)*e^(-(5*x 
^3 + 9*x^2 - 20)/(5*x^2 + 9*x)) + 119*x + 180)*e^((2*x^2*e^(-(5*x^3 + 9*x^ 
2 - 20)/(5*x^2 + 9*x)) + 1)*e^((5*x^3 + 9*x^2 - 20)/(5*x^2 + 9*x))/x + (5* 
x^3 + 9*x^2 - 20)/(5*x^2 + 9*x))/(25*x^5 + 90*x^4 + 81*x^3), x)
 

3.14.7.9 Mupad [B] (verification not implemented)

Time = 19.23 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \begin {dmath*} \int \frac {e^{\frac {e^{-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (1+2 e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} x^2\right )}{x}-\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (180+119 x-9 x^2+65 x^3+25 x^4+e^{\frac {20-9 x^2-5 x^3}{9 x+5 x^2}} \left (162 x^3+180 x^4+50 x^5\right )\right )}{81 x^3+90 x^4+25 x^5} \, dx={\mathrm {e}}^{\frac {{\mathrm {e}}^{\frac {5\,x^2}{5\,x+9}}\,{\mathrm {e}}^{-\frac {20}{5\,x^2+9\,x}}\,{\mathrm {e}}^{\frac {9\,x}{5\,x+9}}}{x}}\,{\mathrm {e}}^{2\,x} \end {dmath*}

input

int((exp((9*x^2 + 5*x^3 - 20)/(9*x + 5*x^2))*exp((exp((9*x^2 + 5*x^3 - 20) 
/(9*x + 5*x^2))*(2*x^2*exp(-(9*x^2 + 5*x^3 - 20)/(9*x + 5*x^2)) + 1))/x)*( 
119*x + exp(-(9*x^2 + 5*x^3 - 20)/(9*x + 5*x^2))*(162*x^3 + 180*x^4 + 50*x 
^5) - 9*x^2 + 65*x^3 + 25*x^4 + 180))/(81*x^3 + 90*x^4 + 25*x^5),x)
 

output

exp((exp((5*x^2)/(5*x + 9))*exp(-20/(9*x + 5*x^2))*exp((9*x)/(5*x + 9)))/x 
)*exp(2*x)