\(\int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} (1944 x-5508 x^2+864 x^3)+e^{\frac {2 (3+x)}{-3+2 x}} (1944 x^3-3564 x^4+864 x^5)+e^{\frac {3+x}{-3+2 x}} (648 x^5-972 x^6+288 x^7)}{729-972 x+324 x^2} \, dx\) [2894]

Optimal result

Integrand size = 135, antiderivative size = 23 \[ \int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} \left (1944 x-5508 x^2+864 x^3\right )+e^{\frac {2 (3+x)}{-3+2 x}} \left (1944 x^3-3564 x^4+864 x^5\right )+e^{\frac {3+x}{-3+2 x}} \left (648 x^5-972 x^6+288 x^7\right )}{729-972 x+324 x^2} \, dx=\left (e^{\frac {3+x}{-3+2 x}}+\frac {x^2}{3}\right )^4 \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} \left (1944 x-5508 x^2+864 x^3\right )+e^{\frac {2 (3+x)}{-3+2 x}} \left (1944 x^3-3564 x^4+864 x^5\right )+e^{\frac {3+x}{-3+2 x}} \left (648 x^5-972 x^6+288 x^7\right )}{729-972 x+324 x^2} \, dx=\frac {1}{81} \left (3 e^{\frac {3+x}{-3+2 x}}+x^2\right )^4 \] Input:

Integrate[(-2916*E^((4*(3 + x))/(-3 + 2*x)) + 72*x^7 - 96*x^8 + 32*x^9 + E 
^((3*(3 + x))/(-3 + 2*x))*(1944*x - 5508*x^2 + 864*x^3) + E^((2*(3 + x))/( 
-3 + 2*x))*(1944*x^3 - 3564*x^4 + 864*x^5) + E^((3 + x)/(-3 + 2*x))*(648*x 
^5 - 972*x^6 + 288*x^7))/(729 - 972*x + 324*x^2),x]
 

Output:

(3*E^((3 + x)/(-3 + 2*x)) + x^2)^4/81
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {7239, 27, 25, 7237}

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.

\(\displaystyle \int \frac {32 x^9-96 x^8+72 x^7+e^{\frac {3 (x+3)}{2 x-3}} \left (864 x^3-5508 x^2+1944 x\right )+e^{\frac {x+3}{2 x-3}} \left (288 x^7-972 x^6+648 x^5\right )+e^{\frac {2 (x+3)}{2 x-3}} \left (864 x^5-3564 x^4+1944 x^3\right )-2916 e^{\frac {4 (x+3)}{2 x-3}}}{324 x^2-972 x+729} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {4 \left (2 (3-2 x)^2 x-27 e^{\frac {x+3}{2 x-3}}\right ) \left (x^2+3 e^{\frac {x+3}{2 x-3}}\right )^3}{81 (3-2 x)^2}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{81} \int -\frac {\left (27 e^{-\frac {x+3}{3-2 x}}-2 (3-2 x)^2 x\right ) \left (x^2+3 e^{-\frac {x+3}{3-2 x}}\right )^3}{(3-2 x)^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {4}{81} \int \frac {\left (27 e^{-\frac {x+3}{3-2 x}}-2 (3-2 x)^2 x\right ) \left (x^2+3 e^{-\frac {x+3}{3-2 x}}\right )^3}{(3-2 x)^2}dx\)

\(\Big \downarrow \) 7237

\(\displaystyle \frac {1}{81} \left (x^2+3 e^{-\frac {x+3}{3-2 x}}\right )^4\)

Input:

Int[(-2916*E^((4*(3 + x))/(-3 + 2*x)) + 72*x^7 - 96*x^8 + 32*x^9 + E^((3*( 
3 + x))/(-3 + 2*x))*(1944*x - 5508*x^2 + 864*x^3) + E^((2*(3 + x))/(-3 + 2 
*x))*(1944*x^3 - 3564*x^4 + 864*x^5) + E^((3 + x)/(-3 + 2*x))*(648*x^5 - 9 
72*x^6 + 288*x^7))/(729 - 972*x + 324*x^2),x]
 

Output:

(3/E^((3 + x)/(3 - 2*x)) + x^2)^4/81
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 7237

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si 
mp[q*(y^(m + 1)/(m + 1)), x] /;  !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
 

rule 7239

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(20)=40\).

Time = 4.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.30

method	result	size
risch	\(\frac {x^{8}}{81}+{\mathrm e}^{\frac {4 x +12}{-3+2 x}}+\frac {4 \,{\mathrm e}^{\frac {3 x +9}{-3+2 x}} x^{2}}{3}+\frac {2 \,{\mathrm e}^{\frac {2 x +6}{-3+2 x}} x^{4}}{3}+\frac {4 \,{\mathrm e}^{\frac {3+x}{-3+2 x}} x^{6}}{27}\)	\(76\)
parallelrisch	\(\frac {x^{8}}{81}+{\mathrm e}^{\frac {4 x +12}{-3+2 x}}+\frac {4 \,{\mathrm e}^{\frac {3 x +9}{-3+2 x}} x^{2}}{3}+\frac {2 \,{\mathrm e}^{\frac {2 x +6}{-3+2 x}} x^{4}}{3}+\frac {4 \,{\mathrm e}^{\frac {3+x}{-3+2 x}} x^{6}}{27}\)	\(76\)
parts	\(\frac {x^{8}}{81}+{\mathrm e}^{2+\frac {18}{-3+2 x}}+\frac {{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{2}}{3}+2 \,{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )+3 \,{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (-3+2 x \right )}}+\frac {9 \,{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )}{2}+\frac {{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )^{4}}{24}+\frac {{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )^{3}}{2}+\frac {9 \,{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )^{2}}{4}+\frac {27 \,{\mathrm e}^{1+\frac {9}{-3+2 x}}}{8}+\frac {27 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )}{8}+\frac {{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{6}}{432}+\frac {{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{5}}{24}+\frac {5 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{4}}{16}+\frac {5 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{3}}{4}+\frac {45 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{2}}{16}+\frac {27 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}}}{16}\)	\(325\)
derivativedivides	\(\frac {27 x}{16}+\frac {45 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{2}}{16}+\frac {5 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{3}}{4}+\frac {5 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{4}}{16}+\frac {{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{5}}{24}+\frac {{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{6}}{432}+\frac {27 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )}{8}+\frac {35 \left (-3+2 x \right )^{4}}{128}+\frac {7 \left (-3+2 x \right )^{5}}{96}+\frac {7 \left (-3+2 x \right )^{6}}{576}+\frac {\left (-3+2 x \right )^{8}}{20736}+\frac {\left (-3+2 x \right )^{7}}{864}+\frac {21 \left (-3+2 x \right )^{3}}{32}-\frac {81}{32}+\frac {27 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}}}{16}+\frac {63 \left (-3+2 x \right )^{2}}{64}+\frac {{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{2}}{3}+2 \,{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )+\frac {9 \,{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )}{2}+\frac {{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )^{4}}{24}+\frac {{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )^{3}}{2}+\frac {9 \,{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )^{2}}{4}+{\mathrm e}^{2+\frac {18}{-3+2 x}}+\frac {27 \,{\mathrm e}^{1+\frac {9}{-3+2 x}}}{8}+3 \,{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (-3+2 x \right )}}\)	\(387\)
default	\(\frac {27 x}{16}+\frac {45 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{2}}{16}+\frac {5 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{3}}{4}+\frac {5 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{4}}{16}+\frac {{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{5}}{24}+\frac {{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{6}}{432}+\frac {27 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )}{8}+\frac {35 \left (-3+2 x \right )^{4}}{128}+\frac {7 \left (-3+2 x \right )^{5}}{96}+\frac {7 \left (-3+2 x \right )^{6}}{576}+\frac {\left (-3+2 x \right )^{8}}{20736}+\frac {\left (-3+2 x \right )^{7}}{864}+\frac {21 \left (-3+2 x \right )^{3}}{32}-\frac {81}{32}+\frac {27 \,{\mathrm e}^{\frac {1}{2}+\frac {9}{2 \left (-3+2 x \right )}}}{16}+\frac {63 \left (-3+2 x \right )^{2}}{64}+\frac {{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )^{2}}{3}+2 \,{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (-3+2 x \right )}} \left (-3+2 x \right )+\frac {9 \,{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )}{2}+\frac {{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )^{4}}{24}+\frac {{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )^{3}}{2}+\frac {9 \,{\mathrm e}^{1+\frac {9}{-3+2 x}} \left (-3+2 x \right )^{2}}{4}+{\mathrm e}^{2+\frac {18}{-3+2 x}}+\frac {27 \,{\mathrm e}^{1+\frac {9}{-3+2 x}}}{8}+3 \,{\mathrm e}^{\frac {3}{2}+\frac {27}{2 \left (-3+2 x \right )}}\)	\(387\)
orering	\(\text {Expression too large to display}\)	\(11447\)

Input:

int((-2916*exp((3+x)/(-3+2*x))^4+(864*x^3-5508*x^2+1944*x)*exp((3+x)/(-3+2 
*x))^3+(864*x^5-3564*x^4+1944*x^3)*exp((3+x)/(-3+2*x))^2+(288*x^7-972*x^6+ 
648*x^5)*exp((3+x)/(-3+2*x))+32*x^9-96*x^8+72*x^7)/(324*x^2-972*x+729),x,m 
ethod=_RETURNVERBOSE)
 

Output:

1/81*x^8+exp((3+x)/(-3+2*x))^4+4/3*exp((3+x)/(-3+2*x))^3*x^2+2/3*exp((3+x) 
/(-3+2*x))^2*x^4+4/27*exp((3+x)/(-3+2*x))*x^6
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (22) = 44\).

Time = 0.10 (sec) , antiderivative size = 72, normalized size of antiderivative = 3.13 \[ \int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} \left (1944 x-5508 x^2+864 x^3\right )+e^{\frac {2 (3+x)}{-3+2 x}} \left (1944 x^3-3564 x^4+864 x^5\right )+e^{\frac {3+x}{-3+2 x}} \left (648 x^5-972 x^6+288 x^7\right )}{729-972 x+324 x^2} \, dx=\frac {1}{81} \, x^{8} + \frac {4}{27} \, x^{6} e^{\left (\frac {x + 3}{2 \, x - 3}\right )} + \frac {2}{3} \, x^{4} e^{\left (\frac {2 \, {\left (x + 3\right )}}{2 \, x - 3}\right )} + \frac {4}{3} \, x^{2} e^{\left (\frac {3 \, {\left (x + 3\right )}}{2 \, x - 3}\right )} + e^{\left (\frac {4 \, {\left (x + 3\right )}}{2 \, x - 3}\right )} \] Input:

integrate((-2916*exp((3+x)/(-3+2*x))^4+(864*x^3-5508*x^2+1944*x)*exp((3+x) 
/(-3+2*x))^3+(864*x^5-3564*x^4+1944*x^3)*exp((3+x)/(-3+2*x))^2+(288*x^7-97 
2*x^6+648*x^5)*exp((3+x)/(-3+2*x))+32*x^9-96*x^8+72*x^7)/(324*x^2-972*x+72 
9),x, algorithm="fricas")
 

Output:

1/81*x^8 + 4/27*x^6*e^((x + 3)/(2*x - 3)) + 2/3*x^4*e^(2*(x + 3)/(2*x - 3) 
) + 4/3*x^2*e^(3*(x + 3)/(2*x - 3)) + e^(4*(x + 3)/(2*x - 3))
                                                                                    
                                                                                    
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (15) = 30\).

Time = 0.15 (sec) , antiderivative size = 70, normalized size of antiderivative = 3.04 \[ \int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} \left (1944 x-5508 x^2+864 x^3\right )+e^{\frac {2 (3+x)}{-3+2 x}} \left (1944 x^3-3564 x^4+864 x^5\right )+e^{\frac {3+x}{-3+2 x}} \left (648 x^5-972 x^6+288 x^7\right )}{729-972 x+324 x^2} \, dx=\frac {x^{8}}{81} + \frac {4 x^{6} e^{\frac {x + 3}{2 x - 3}}}{27} + \frac {2 x^{4} e^{\frac {2 \left (x + 3\right )}{2 x - 3}}}{3} + \frac {4 x^{2} e^{\frac {3 \left (x + 3\right )}{2 x - 3}}}{3} + e^{\frac {4 \left (x + 3\right )}{2 x - 3}} \] Input:

integrate((-2916*exp((3+x)/(-3+2*x))**4+(864*x**3-5508*x**2+1944*x)*exp((3 
+x)/(-3+2*x))**3+(864*x**5-3564*x**4+1944*x**3)*exp((3+x)/(-3+2*x))**2+(28 
8*x**7-972*x**6+648*x**5)*exp((3+x)/(-3+2*x))+32*x**9-96*x**8+72*x**7)/(32 
4*x**2-972*x+729),x)
 

Output:

x**8/81 + 4*x**6*exp((x + 3)/(2*x - 3))/27 + 2*x**4*exp(2*(x + 3)/(2*x - 3 
))/3 + 4*x**2*exp(3*(x + 3)/(2*x - 3))/3 + exp(4*(x + 3)/(2*x - 3))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (22) = 44\).

Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.00 \[ \int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} \left (1944 x-5508 x^2+864 x^3\right )+e^{\frac {2 (3+x)}{-3+2 x}} \left (1944 x^3-3564 x^4+864 x^5\right )+e^{\frac {3+x}{-3+2 x}} \left (648 x^5-972 x^6+288 x^7\right )}{729-972 x+324 x^2} \, dx=\frac {1}{81} \, x^{8} + \frac {4}{27} \, x^{6} e^{\left (\frac {9}{2 \, {\left (2 \, x - 3\right )}} + \frac {1}{2}\right )} + \frac {2}{3} \, x^{4} e^{\left (\frac {9}{2 \, x - 3} + 1\right )} + \frac {4}{3} \, x^{2} e^{\left (\frac {27}{2 \, {\left (2 \, x - 3\right )}} + \frac {3}{2}\right )} + e^{\left (\frac {18}{2 \, x - 3} + 2\right )} \] Input:

integrate((-2916*exp((3+x)/(-3+2*x))^4+(864*x^3-5508*x^2+1944*x)*exp((3+x) 
/(-3+2*x))^3+(864*x^5-3564*x^4+1944*x^3)*exp((3+x)/(-3+2*x))^2+(288*x^7-97 
2*x^6+648*x^5)*exp((3+x)/(-3+2*x))+32*x^9-96*x^8+72*x^7)/(324*x^2-972*x+72 
9),x, algorithm="maxima")
 

Output:

1/81*x^8 + 4/27*x^6*e^(9/2/(2*x - 3) + 1/2) + 2/3*x^4*e^(9/(2*x - 3) + 1) 
+ 4/3*x^2*e^(27/2/(2*x - 3) + 3/2) + e^(18/(2*x - 3) + 2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 85 vs. \(2 (22) = 44\).

Time = 0.14 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.70 \[ \int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} \left (1944 x-5508 x^2+864 x^3\right )+e^{\frac {2 (3+x)}{-3+2 x}} \left (1944 x^3-3564 x^4+864 x^5\right )+e^{\frac {3+x}{-3+2 x}} \left (648 x^5-972 x^6+288 x^7\right )}{729-972 x+324 x^2} \, dx=\frac {1}{81} \, {\left (x^{8} e^{3} + 12 \, x^{6} e^{\left (\frac {3 \, x}{2 \, x - 3} + 2\right )} + 54 \, x^{4} e^{\left (\frac {6 \, x}{2 \, x - 3} + 1\right )} + 108 \, x^{2} e^{\left (\frac {9 \, x}{2 \, x - 3}\right )}\right )} e^{\left (-3\right )} + e^{\left (\frac {4 \, x}{2 \, x - 3} + \frac {12}{2 \, x - 3}\right )} \] Input:

integrate((-2916*exp((3+x)/(-3+2*x))^4+(864*x^3-5508*x^2+1944*x)*exp((3+x) 
/(-3+2*x))^3+(864*x^5-3564*x^4+1944*x^3)*exp((3+x)/(-3+2*x))^2+(288*x^7-97 
2*x^6+648*x^5)*exp((3+x)/(-3+2*x))+32*x^9-96*x^8+72*x^7)/(324*x^2-972*x+72 
9),x, algorithm="giac")
 

Output:

1/81*(x^8*e^3 + 12*x^6*e^(3*x/(2*x - 3) + 2) + 54*x^4*e^(6*x/(2*x - 3) + 1 
) + 108*x^2*e^(9*x/(2*x - 3)))*e^(-3) + e^(4*x/(2*x - 3) + 12/(2*x - 3))
 

Mupad [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.52 \[ \int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} \left (1944 x-5508 x^2+864 x^3\right )+e^{\frac {2 (3+x)}{-3+2 x}} \left (1944 x^3-3564 x^4+864 x^5\right )+e^{\frac {3+x}{-3+2 x}} \left (648 x^5-972 x^6+288 x^7\right )}{729-972 x+324 x^2} \, dx={\mathrm {e}}^{\frac {4\,x}{2\,x-3}+\frac {12}{2\,x-3}}+\frac {4\,x^6\,{\mathrm {e}}^{\frac {x}{2\,x-3}+\frac {3}{2\,x-3}}}{27}+\frac {2\,x^4\,{\mathrm {e}}^{\frac {2\,x}{2\,x-3}+\frac {6}{2\,x-3}}}{3}+\frac {4\,x^2\,{\mathrm {e}}^{\frac {3\,x}{2\,x-3}+\frac {9}{2\,x-3}}}{3}+\frac {x^8}{81} \] Input:

int((exp((3*(x + 3))/(2*x - 3))*(1944*x - 5508*x^2 + 864*x^3) - 2916*exp(( 
4*(x + 3))/(2*x - 3)) + exp((x + 3)/(2*x - 3))*(648*x^5 - 972*x^6 + 288*x^ 
7) + exp((2*(x + 3))/(2*x - 3))*(1944*x^3 - 3564*x^4 + 864*x^5) + 72*x^7 - 
 96*x^8 + 32*x^9)/(324*x^2 - 972*x + 729),x)
 

Output:

exp((4*x)/(2*x - 3) + 12/(2*x - 3)) + (4*x^6*exp(x/(2*x - 3) + 3/(2*x - 3) 
))/27 + (2*x^4*exp((2*x)/(2*x - 3) + 6/(2*x - 3)))/3 + (4*x^2*exp((3*x)/(2 
*x - 3) + 9/(2*x - 3)))/3 + x^8/81
                                                                                    
                                                                                    
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.26 \[ \int \frac {-2916 e^{\frac {4 (3+x)}{-3+2 x}}+72 x^7-96 x^8+32 x^9+e^{\frac {3 (3+x)}{-3+2 x}} \left (1944 x-5508 x^2+864 x^3\right )+e^{\frac {2 (3+x)}{-3+2 x}} \left (1944 x^3-3564 x^4+864 x^5\right )+e^{\frac {3+x}{-3+2 x}} \left (648 x^5-972 x^6+288 x^7\right )}{729-972 x+324 x^2} \, dx=\frac {4 e^{\frac {x +12}{2 x -3}} e \,x^{2}}{3}+\frac {4 e^{\frac {x +3}{2 x -3}} x^{6}}{27}+e^{\frac {18}{2 x -3}} e^{2}+\frac {2 e^{\frac {9}{2 x -3}} e \,x^{4}}{3}+\frac {x^{8}}{81} \] Input:

int((-2916*exp((3+x)/(-3+2*x))^4+(864*x^3-5508*x^2+1944*x)*exp((3+x)/(-3+2 
*x))^3+(864*x^5-3564*x^4+1944*x^3)*exp((3+x)/(-3+2*x))^2+(288*x^7-972*x^6+ 
648*x^5)*exp((3+x)/(-3+2*x))+32*x^9-96*x^8+72*x^7)/(324*x^2-972*x+729),x)
 

Output:

(108*e**((x + 12)/(2*x - 3))*e*x**2 + 12*e**((x + 3)/(2*x - 3))*x**6 + 81* 
e**(18/(2*x - 3))*e**2 + 54*e**(9/(2*x - 3))*e*x**4 + x**8)/81