\(\int \frac {e^{e^{\frac {1}{2} (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2)}+\frac {1}{2} (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2)} (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} (-729 x-1215 x^2-324 x^3)+e^{\frac {x}{3+2 x}} (4374 x+6561 x^2+1944 x^3))}{9+12 x+4 x^2} \, dx\) [126]

Optimal result

Integrand size = 171, antiderivative size = 29 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=e^{e^{\frac {1}{2} x \left (x-81 \left (-3+e^{\frac {x}{3+2 x}}\right )^2 x\right )}} \] Output:

exp(exp(1/2*x*(x-9*x*(exp(x/(3+2*x))-3)*(9*exp(x/(3+2*x))-27))))
 

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.38 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=e^{e^{-\frac {1}{2} \left (728-486 e^{\frac {x}{3+2 x}}+81 e^{\frac {2 x}{3+2 x}}\right ) x^2}} \] Input:

Integrate[(E^(E^((-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2* 
x))*x^2)/2) + (-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x)) 
*x^2)/2)*(-6552*x - 8736*x^2 - 2912*x^3 + E^((2*x)/(3 + 2*x))*(-729*x - 12 
15*x^2 - 324*x^3) + E^(x/(3 + 2*x))*(4374*x + 6561*x^2 + 1944*x^3)))/(9 + 
12*x + 4*x^2),x]
 

Output:

E^E^(-1/2*((728 - 486*E^(x/(3 + 2*x)) + 81*E^((2*x)/(3 + 2*x)))*x^2))
 

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.

\(\displaystyle \int \frac {\left (-2912 x^3-8736 x^2+e^{\frac {2 x}{2 x+3}} \left (-324 x^3-1215 x^2-729 x\right )+e^{\frac {x}{2 x+3}} \left (1944 x^3+6561 x^2+4374 x\right )-6552 x\right ) \exp \left (\exp \left (\frac {1}{2} \left (486 e^{\frac {x}{2 x+3}} x^2-81 e^{\frac {2 x}{2 x+3}} x^2-728 x^2\right )\right )+\frac {1}{2} \left (486 e^{\frac {x}{2 x+3}} x^2-81 e^{\frac {2 x}{2 x+3}} x^2-728 x^2\right )\right )}{4 x^2+12 x+9} \, dx\)

\(\Big \downarrow \) 2007

\(\displaystyle \int \frac {\left (-2912 x^3-8736 x^2+e^{\frac {2 x}{2 x+3}} \left (-324 x^3-1215 x^2-729 x\right )+e^{\frac {x}{2 x+3}} \left (1944 x^3+6561 x^2+4374 x\right )-6552 x\right ) \exp \left (\exp \left (\frac {1}{2} \left (486 e^{\frac {x}{2 x+3}} x^2-81 e^{\frac {2 x}{2 x+3}} x^2-728 x^2\right )\right )+\frac {1}{2} \left (486 e^{\frac {x}{2 x+3}} x^2-81 e^{\frac {2 x}{2 x+3}} x^2-728 x^2\right )\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (\frac {1}{2} \left (486 e^{\frac {x}{2 x+3}} x^2-81 e^{\frac {2 x}{2 x+3}} x^2-728 x^2\right )\right )+\frac {1}{2} \left (486 e^{\frac {x}{2 x+3}} x^2-81 e^{\frac {2 x}{2 x+3}} x^2-728 x^2\right )\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (\frac {1}{2} \left (486 e^{\frac {x}{2 x+3}} x^2-81 e^{\frac {2 x}{2 x+3}} x^2-728 x^2\right )\right )+\frac {1}{2} \left (486 e^{\frac {x}{2 x+3}} x^2-81 e^{\frac {2 x}{2 x+3}} x^2-728 x^2\right )+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (\frac {1}{2} \left (486 e^{\frac {x}{2 x+3}} x^2-81 e^{\frac {2 x}{2 x+3}} x^2-728 x^2\right )\right )+\frac {1}{2} \left (486 e^{\frac {x}{2 x+3}} x^2-81 e^{\frac {2 x}{2 x+3}} x^2-728 x^2\right )+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {x \left (-81 e^{\frac {2 x}{2 x+3}} \left (4 x^2+15 x+9\right )+243 e^{\frac {x}{2 x+3}} \left (8 x^2+27 x+18\right )-728 (2 x+3)^2\right ) \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )}{(2 x+3)^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-728 x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2\right )-\frac {81 (x+3) (4 x+3) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {2 x}{2 x+3}\right )}{(2 x+3)^2}+\frac {243 \left (8 x^2+27 x+18\right ) x \exp \left (\exp \left (-\frac {1}{2} \left (-486 e^{\frac {x}{2 x+3}}+81 e^{\frac {2 x}{2 x+3}}+728\right ) x^2\right )+243 e^{\frac {x}{2 x+3}} x^2-\frac {81}{2} e^{\frac {2 x}{2 x+3}} x^2-364 x^2+\frac {x}{2 x+3}\right )}{(2 x+3)^2}\right )dx\)

Input:

Int[(E^(E^((-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^ 
2)/2) + (-728*x^2 + 486*E^(x/(3 + 2*x))*x^2 - 81*E^((2*x)/(3 + 2*x))*x^2)/ 
2)*(-6552*x - 8736*x^2 - 2912*x^3 + E^((2*x)/(3 + 2*x))*(-729*x - 1215*x^2 
 - 324*x^3) + E^(x/(3 + 2*x))*(4374*x + 6561*x^2 + 1944*x^3)))/(9 + 12*x + 
 4*x^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 3.96 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.21

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {x^{2} \left (-81 \,{\mathrm e}^{\frac {2 x}{3+2 x}}+486 \,{\mathrm e}^{\frac {x}{3+2 x}}-728\right )}{2}}}\)	\(35\)
parallelrisch	\({\mathrm e}^{{\mathrm e}^{-\frac {x^{2} \left (81 \,{\mathrm e}^{\frac {2 x}{3+2 x}}-486 \,{\mathrm e}^{\frac {x}{3+2 x}}+728\right )}{2}}}\)	\(36\)

Input:

int(((-324*x^3-1215*x^2-729*x)*exp(x/(3+2*x))^2+(1944*x^3+6561*x^2+4374*x) 
*exp(x/(3+2*x))-2912*x^3-8736*x^2-6552*x)*exp(-81/2*x^2*exp(x/(3+2*x))^2+2 
43*x^2*exp(x/(3+2*x))-364*x^2)*exp(exp(-81/2*x^2*exp(x/(3+2*x))^2+243*x^2* 
exp(x/(3+2*x))-364*x^2))/(4*x^2+12*x+9),x,method=_RETURNVERBOSE)
 

Output:

exp(exp(1/2*x^2*(-81*exp(2*x/(3+2*x))+486*exp(x/(3+2*x))-728)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=e^{\left (e^{\left (-\frac {81}{2} \, x^{2} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} + 243 \, x^{2} e^{\left (\frac {x}{2 \, x + 3}\right )} - 364 \, x^{2}\right )}\right )} \] Input:

integrate(((-324*x^3-1215*x^2-729*x)*exp(x/(3+2*x))^2+(1944*x^3+6561*x^2+4 
374*x)*exp(x/(3+2*x))-2912*x^3-8736*x^2-6552*x)*exp(-81/2*x^2*exp(x/(3+2*x 
))^2+243*x^2*exp(x/(3+2*x))-364*x^2)*exp(exp(-81/2*x^2*exp(x/(3+2*x))^2+24 
3*x^2*exp(x/(3+2*x))-364*x^2))/(4*x^2+12*x+9),x, algorithm="fricas")
 

Output:

e^(e^(-81/2*x^2*e^(2*x/(2*x + 3)) + 243*x^2*e^(x/(2*x + 3)) - 364*x^2))
 

Sympy [A] (verification not implemented)

Time = 2.52 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=e^{e^{- \frac {81 x^{2} e^{\frac {2 x}{2 x + 3}}}{2} + 243 x^{2} e^{\frac {x}{2 x + 3}} - 364 x^{2}}} \] Input:

integrate(((-324*x**3-1215*x**2-729*x)*exp(x/(3+2*x))**2+(1944*x**3+6561*x 
**2+4374*x)*exp(x/(3+2*x))-2912*x**3-8736*x**2-6552*x)*exp(-81/2*x**2*exp( 
x/(3+2*x))**2+243*x**2*exp(x/(3+2*x))-364*x**2)*exp(exp(-81/2*x**2*exp(x/( 
3+2*x))**2+243*x**2*exp(x/(3+2*x))-364*x**2))/(4*x**2+12*x+9),x)
 

Output:

exp(exp(-81*x**2*exp(2*x/(2*x + 3))/2 + 243*x**2*exp(x/(2*x + 3)) - 364*x* 
*2))
 

Maxima [A] (verification not implemented)

Time = 0.40 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.45 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=e^{\left (e^{\left (243 \, x^{2} e^{\left (-\frac {3}{2 \, {\left (2 \, x + 3\right )}} + \frac {1}{2}\right )} - \frac {81}{2} \, x^{2} e^{\left (-\frac {3}{2 \, x + 3} + 1\right )} - 364 \, x^{2}\right )}\right )} \] Input:

integrate(((-324*x^3-1215*x^2-729*x)*exp(x/(3+2*x))^2+(1944*x^3+6561*x^2+4 
374*x)*exp(x/(3+2*x))-2912*x^3-8736*x^2-6552*x)*exp(-81/2*x^2*exp(x/(3+2*x 
))^2+243*x^2*exp(x/(3+2*x))-364*x^2)*exp(exp(-81/2*x^2*exp(x/(3+2*x))^2+24 
3*x^2*exp(x/(3+2*x))-364*x^2))/(4*x^2+12*x+9),x, algorithm="maxima")
 

Output:

e^(e^(243*x^2*e^(-3/2/(2*x + 3) + 1/2) - 81/2*x^2*e^(-3/(2*x + 3) + 1) - 3 
64*x^2))
 

Giac [F]

\[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=\int { -\frac {{\left (2912 \, x^{3} + 8736 \, x^{2} + 81 \, {\left (4 \, x^{3} + 15 \, x^{2} + 9 \, x\right )} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} - 243 \, {\left (8 \, x^{3} + 27 \, x^{2} + 18 \, x\right )} e^{\left (\frac {x}{2 \, x + 3}\right )} + 6552 \, x\right )} e^{\left (-\frac {81}{2} \, x^{2} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} + 243 \, x^{2} e^{\left (\frac {x}{2 \, x + 3}\right )} - 364 \, x^{2} + e^{\left (-\frac {81}{2} \, x^{2} e^{\left (\frac {2 \, x}{2 \, x + 3}\right )} + 243 \, x^{2} e^{\left (\frac {x}{2 \, x + 3}\right )} - 364 \, x^{2}\right )}\right )}}{4 \, x^{2} + 12 \, x + 9} \,d x } \] Input:

integrate(((-324*x^3-1215*x^2-729*x)*exp(x/(3+2*x))^2+(1944*x^3+6561*x^2+4 
374*x)*exp(x/(3+2*x))-2912*x^3-8736*x^2-6552*x)*exp(-81/2*x^2*exp(x/(3+2*x 
))^2+243*x^2*exp(x/(3+2*x))-364*x^2)*exp(exp(-81/2*x^2*exp(x/(3+2*x))^2+24 
3*x^2*exp(x/(3+2*x))-364*x^2))/(4*x^2+12*x+9),x, algorithm="giac")
 

Output:

integrate(-(2912*x^3 + 8736*x^2 + 81*(4*x^3 + 15*x^2 + 9*x)*e^(2*x/(2*x + 
3)) - 243*(8*x^3 + 27*x^2 + 18*x)*e^(x/(2*x + 3)) + 6552*x)*e^(-81/2*x^2*e 
^(2*x/(2*x + 3)) + 243*x^2*e^(x/(2*x + 3)) - 364*x^2 + e^(-81/2*x^2*e^(2*x 
/(2*x + 3)) + 243*x^2*e^(x/(2*x + 3)) - 364*x^2))/(4*x^2 + 12*x + 9), x)
 

Mupad [B] (verification not implemented)

Time = 3.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.41 \[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx={\mathrm {e}}^{{\mathrm {e}}^{-\frac {81\,x^2\,{\mathrm {e}}^{\frac {2\,x}{2\,x+3}}}{2}}\,{\mathrm {e}}^{243\,x^2\,{\mathrm {e}}^{\frac {x}{2\,x+3}}}\,{\mathrm {e}}^{-364\,x^2}} \] Input:

int(-(exp(243*x^2*exp(x/(2*x + 3)) - (81*x^2*exp((2*x)/(2*x + 3)))/2 - 364 
*x^2)*exp(exp(243*x^2*exp(x/(2*x + 3)) - (81*x^2*exp((2*x)/(2*x + 3)))/2 - 
 364*x^2))*(6552*x + exp((2*x)/(2*x + 3))*(729*x + 1215*x^2 + 324*x^3) - e 
xp(x/(2*x + 3))*(4374*x + 6561*x^2 + 1944*x^3) + 8736*x^2 + 2912*x^3))/(12 
*x + 4*x^2 + 9),x)
 

Output:

exp(exp(-(81*x^2*exp((2*x)/(2*x + 3)))/2)*exp(243*x^2*exp(x/(2*x + 3)))*ex 
p(-364*x^2))
 

Reduce [F]

\[ \int \frac {e^{e^{\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )}+\frac {1}{2} \left (-728 x^2+486 e^{\frac {x}{3+2 x}} x^2-81 e^{\frac {2 x}{3+2 x}} x^2\right )} \left (-6552 x-8736 x^2-2912 x^3+e^{\frac {2 x}{3+2 x}} \left (-729 x-1215 x^2-324 x^3\right )+e^{\frac {x}{3+2 x}} \left (4374 x+6561 x^2+1944 x^3\right )\right )}{9+12 x+4 x^2} \, dx=\text {too large to display} \] Input:

int(((-324*x^3-1215*x^2-729*x)*exp(x/(3+2*x))^2+(1944*x^3+6561*x^2+4374*x) 
*exp(x/(3+2*x))-2912*x^3-8736*x^2-6552*x)*exp(-81/2*x^2*exp(x/(3+2*x))^2+2 
43*x^2*exp(x/(3+2*x))-364*x^2)*exp(exp(-81/2*x^2*exp(x/(3+2*x))^2+243*x^2* 
exp(x/(3+2*x))-364*x^2))/(4*x^2+12*x+9),x)
 

Output:

1944*int((e**((486*e**((1456*e**(3/(2*x + 3))*x**3 + 2184*e**(3/(2*x + 3)) 
*x**2 + 2*e**(3/(2*x + 3))*x + 162*e*x**3 + 243*e*x**2)/(4*e**(3/(2*x + 3) 
)*x + 6*e**(3/(2*x + 3))))*x**3 + 729*e**((1456*e**(3/(2*x + 3))*x**3 + 21 
84*e**(3/(2*x + 3))*x**2 + 2*e**(3/(2*x + 3))*x + 162*e*x**3 + 243*e*x**2) 
/(4*e**(3/(2*x + 3))*x + 6*e**(3/(2*x + 3))))*x**2 + e**((728*e**(3/(2*x + 
 3))*x**2 + 81*e*x**2)/(2*e**(3/(2*x + 3))))*x + 2*e**(243*e**(x/(2*x + 3) 
)*x**2)*x + 3*e**(243*e**(x/(2*x + 3))*x**2))/(2*e**((728*e**(3/(2*x + 3)) 
*x**2 + 81*e*x**2)/(2*e**(3/(2*x + 3))))*x + 3*e**((728*e**(3/(2*x + 3))*x 
**2 + 81*e*x**2)/(2*e**(3/(2*x + 3))))))*x**3)/(4*e**((728*e**(3/(2*x + 3) 
)*x**2 + 81*e*x**2)/(2*e**(3/(2*x + 3))))*x**2 + 12*e**((728*e**(3/(2*x + 
3))*x**2 + 81*e*x**2)/(2*e**(3/(2*x + 3))))*x + 9*e**((728*e**(3/(2*x + 3) 
)*x**2 + 81*e*x**2)/(2*e**(3/(2*x + 3))))),x) + 6561*int((e**((486*e**((14 
56*e**(3/(2*x + 3))*x**3 + 2184*e**(3/(2*x + 3))*x**2 + 2*e**(3/(2*x + 3)) 
*x + 162*e*x**3 + 243*e*x**2)/(4*e**(3/(2*x + 3))*x + 6*e**(3/(2*x + 3)))) 
*x**3 + 729*e**((1456*e**(3/(2*x + 3))*x**3 + 2184*e**(3/(2*x + 3))*x**2 + 
 2*e**(3/(2*x + 3))*x + 162*e*x**3 + 243*e*x**2)/(4*e**(3/(2*x + 3))*x + 6 
*e**(3/(2*x + 3))))*x**2 + e**((728*e**(3/(2*x + 3))*x**2 + 81*e*x**2)/(2* 
e**(3/(2*x + 3))))*x + 2*e**(243*e**(x/(2*x + 3))*x**2)*x + 3*e**(243*e**( 
x/(2*x + 3))*x**2))/(2*e**((728*e**(3/(2*x + 3))*x**2 + 81*e*x**2)/(2*e**( 
3/(2*x + 3))))*x + 3*e**((728*e**(3/(2*x + 3))*x**2 + 81*e*x**2)/(2*e**...