\(\int \frac {160+e^x (96-64 x)+32 x-64 x^2+(192 x-128 x^2) \log (x)}{-3 x^3+2 x^4+e^{3 x} (-3+2 x)+e^{2 x} (-9 x+6 x^2)+e^x (-9 x^2+6 x^3)+(-3 x^6+2 x^7) \log ^3(x)+(-9 x^2+6 x^3+e^{2 x} (-9+6 x)+e^x (-18 x+12 x^2)) \log (\frac {1}{-3+2 x})+(-9 x+6 x^2+e^x (-9+6 x)) \log ^2(\frac {1}{-3+2 x})+(-3+2 x) \log ^3(\frac {1}{-3+2 x})+\log ^2(x) (-9 x^5+6 x^6+e^x (-9 x^4+6 x^5)+(-9 x^4+6 x^5) \log (\frac {1}{-3+2 x}))+\log (x) (-9 x^4+6 x^5+e^{2 x} (-9 x^2+6 x^3)+e^x (-18 x^3+12 x^4)+(-18 x^3+12 x^4+e^x (-18 x^2+12 x^3)) \log (\frac {1}{-3+2 x})+(-9 x^2+6 x^3) \log ^2(\frac {1}{-3+2 x}))} \, dx\) [2483]

Optimal result

Integrand size = 345, antiderivative size = 23 \[ \int \frac {160+e^x (96-64 x)+32 x-64 x^2+\left (192 x-128 x^2\right ) \log (x)}{-3 x^3+2 x^4+e^{3 x} (-3+2 x)+e^{2 x} \left (-9 x+6 x^2\right )+e^x \left (-9 x^2+6 x^3\right )+\left (-3 x^6+2 x^7\right ) \log ^3(x)+\left (-9 x^2+6 x^3+e^{2 x} (-9+6 x)+e^x \left (-18 x+12 x^2\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x+6 x^2+e^x (-9+6 x)\right ) \log ^2\left (\frac {1}{-3+2 x}\right )+(-3+2 x) \log ^3\left (\frac {1}{-3+2 x}\right )+\log ^2(x) \left (-9 x^5+6 x^6+e^x \left (-9 x^4+6 x^5\right )+\left (-9 x^4+6 x^5\right ) \log \left (\frac {1}{-3+2 x}\right )\right )+\log (x) \left (-9 x^4+6 x^5+e^{2 x} \left (-9 x^2+6 x^3\right )+e^x \left (-18 x^3+12 x^4\right )+\left (-18 x^3+12 x^4+e^x \left (-18 x^2+12 x^3\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x^2+6 x^3\right ) \log ^2\left (\frac {1}{-3+2 x}\right )\right )} \, dx=\frac {16}{\left (e^x+x+x^2 \log (x)+\log \left (\frac {1}{-3+2 x}\right )\right )^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {160+e^x (96-64 x)+32 x-64 x^2+\left (192 x-128 x^2\right ) \log (x)}{-3 x^3+2 x^4+e^{3 x} (-3+2 x)+e^{2 x} \left (-9 x+6 x^2\right )+e^x \left (-9 x^2+6 x^3\right )+\left (-3 x^6+2 x^7\right ) \log ^3(x)+\left (-9 x^2+6 x^3+e^{2 x} (-9+6 x)+e^x \left (-18 x+12 x^2\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x+6 x^2+e^x (-9+6 x)\right ) \log ^2\left (\frac {1}{-3+2 x}\right )+(-3+2 x) \log ^3\left (\frac {1}{-3+2 x}\right )+\log ^2(x) \left (-9 x^5+6 x^6+e^x \left (-9 x^4+6 x^5\right )+\left (-9 x^4+6 x^5\right ) \log \left (\frac {1}{-3+2 x}\right )\right )+\log (x) \left (-9 x^4+6 x^5+e^{2 x} \left (-9 x^2+6 x^3\right )+e^x \left (-18 x^3+12 x^4\right )+\left (-18 x^3+12 x^4+e^x \left (-18 x^2+12 x^3\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x^2+6 x^3\right ) \log ^2\left (\frac {1}{-3+2 x}\right )\right )} \, dx=\frac {16}{\left (e^x+x+x^2 \log (x)+\log \left (\frac {1}{-3+2 x}\right )\right )^2} \] Input:

Integrate[(160 + E^x*(96 - 64*x) + 32*x - 64*x^2 + (192*x - 128*x^2)*Log[x 
])/(-3*x^3 + 2*x^4 + E^(3*x)*(-3 + 2*x) + E^(2*x)*(-9*x + 6*x^2) + E^x*(-9 
*x^2 + 6*x^3) + (-3*x^6 + 2*x^7)*Log[x]^3 + (-9*x^2 + 6*x^3 + E^(2*x)*(-9 
+ 6*x) + E^x*(-18*x + 12*x^2))*Log[(-3 + 2*x)^(-1)] + (-9*x + 6*x^2 + E^x* 
(-9 + 6*x))*Log[(-3 + 2*x)^(-1)]^2 + (-3 + 2*x)*Log[(-3 + 2*x)^(-1)]^3 + L 
og[x]^2*(-9*x^5 + 6*x^6 + E^x*(-9*x^4 + 6*x^5) + (-9*x^4 + 6*x^5)*Log[(-3 
+ 2*x)^(-1)]) + Log[x]*(-9*x^4 + 6*x^5 + E^(2*x)*(-9*x^2 + 6*x^3) + E^x*(- 
18*x^3 + 12*x^4) + (-18*x^3 + 12*x^4 + E^x*(-18*x^2 + 12*x^3))*Log[(-3 + 2 
*x)^(-1)] + (-9*x^2 + 6*x^3)*Log[(-3 + 2*x)^(-1)]^2)),x]
 

Output:

16/(E^x + x + x^2*Log[x] + Log[(-3 + 2*x)^(-1)])^2
 

Rubi [A] (verified)

Time = 1.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, 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.

\(\displaystyle \int \frac {-64 x^2+\left (192 x-128 x^2\right ) \log (x)+32 x+e^x (96-64 x)+160}{2 x^4-3 x^3+e^{2 x} \left (6 x^2-9 x\right )+\left (6 x^2-9 x+e^x (6 x-9)\right ) \log ^2\left (\frac {1}{2 x-3}\right )+\left (2 x^7-3 x^6\right ) \log ^3(x)+e^x \left (6 x^3-9 x^2\right )+\left (6 x^3-9 x^2+e^x \left (12 x^2-18 x\right )+e^{2 x} (6 x-9)\right ) \log \left (\frac {1}{2 x-3}\right )+\log ^2(x) \left (6 x^6-9 x^5+e^x \left (6 x^5-9 x^4\right )+\left (6 x^5-9 x^4\right ) \log \left (\frac {1}{2 x-3}\right )\right )+\log (x) \left (6 x^5-9 x^4+e^x \left (12 x^4-18 x^3\right )+e^{2 x} \left (6 x^3-9 x^2\right )+\left (6 x^3-9 x^2\right ) \log ^2\left (\frac {1}{2 x-3}\right )+\left (12 x^4-18 x^3+e^x \left (12 x^3-18 x^2\right )\right ) \log \left (\frac {1}{2 x-3}\right )\right )+e^{3 x} (2 x-3)+(2 x-3) \log ^3\left (\frac {1}{2 x-3}\right )} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {32 \left (2 x^2-x+e^x (2 x-3)+2 (2 x-3) x \log (x)-5\right )}{(3-2 x) \left (x^2 \log (x)+x+e^x+\log \left (\frac {1}{2 x-3}\right )\right )^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 32 \int -\frac {-2 x^2+2 (3-2 x) \log (x) x+x+e^x (3-2 x)+5}{(3-2 x) \left (\log (x) x^2+x+e^x+\log \left (\frac {1}{2 x-3}\right )\right )^3}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -32 \int \frac {-2 x^2+2 (3-2 x) \log (x) x+x+e^x (3-2 x)+5}{(3-2 x) \left (\log (x) x^2+x+e^x+\log \left (\frac {1}{2 x-3}\right )\right )^3}dx\)

\(\Big \downarrow \) 7237

\(\displaystyle \frac {16}{\left (x^2 \log (x)+x+e^x+\log \left (\frac {1}{2 x-3}\right )\right )^2}\)

Input:

Int[(160 + E^x*(96 - 64*x) + 32*x - 64*x^2 + (192*x - 128*x^2)*Log[x])/(-3 
*x^3 + 2*x^4 + E^(3*x)*(-3 + 2*x) + E^(2*x)*(-9*x + 6*x^2) + E^x*(-9*x^2 + 
 6*x^3) + (-3*x^6 + 2*x^7)*Log[x]^3 + (-9*x^2 + 6*x^3 + E^(2*x)*(-9 + 6*x) 
 + E^x*(-18*x + 12*x^2))*Log[(-3 + 2*x)^(-1)] + (-9*x + 6*x^2 + E^x*(-9 + 
6*x))*Log[(-3 + 2*x)^(-1)]^2 + (-3 + 2*x)*Log[(-3 + 2*x)^(-1)]^3 + Log[x]^ 
2*(-9*x^5 + 6*x^6 + E^x*(-9*x^4 + 6*x^5) + (-9*x^4 + 6*x^5)*Log[(-3 + 2*x) 
^(-1)]) + Log[x]*(-9*x^4 + 6*x^5 + E^(2*x)*(-9*x^2 + 6*x^3) + E^x*(-18*x^3 
 + 12*x^4) + (-18*x^3 + 12*x^4 + E^x*(-18*x^2 + 12*x^3))*Log[(-3 + 2*x)^(- 
1)] + (-9*x^2 + 6*x^3)*Log[(-3 + 2*x)^(-1)]^2)),x]
 

Output:

16/(E^x + x + x^2*Log[x] + Log[(-3 + 2*x)^(-1)])^2
 

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 [C] (verified)

Result contains complex when optimal does not.

Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52

\[-\frac {64}{\left (2 i x^{2} \ln \left (x \right )-2 i \ln \left (2\right )+2 i x +2 i {\mathrm e}^{x}-2 i \ln \left (x -\frac {3}{2}\right )\right )^{2}}\]

Input:

int(((-128*x^2+192*x)*ln(x)+(-64*x+96)*exp(x)-64*x^2+32*x+160)/((2*x^7-3*x 
^6)*ln(x)^3+((6*x^5-9*x^4)*ln(1/(-3+2*x))+(6*x^5-9*x^4)*exp(x)+6*x^6-9*x^5 
)*ln(x)^2+((6*x^3-9*x^2)*ln(1/(-3+2*x))^2+((12*x^3-18*x^2)*exp(x)+12*x^4-1 
8*x^3)*ln(1/(-3+2*x))+(6*x^3-9*x^2)*exp(x)^2+(12*x^4-18*x^3)*exp(x)+6*x^5- 
9*x^4)*ln(x)+(-3+2*x)*ln(1/(-3+2*x))^3+((6*x-9)*exp(x)+6*x^2-9*x)*ln(1/(-3 
+2*x))^2+((6*x-9)*exp(x)^2+(12*x^2-18*x)*exp(x)+6*x^3-9*x^2)*ln(1/(-3+2*x) 
)+(-3+2*x)*exp(x)^3+(6*x^2-9*x)*exp(x)^2+(6*x^3-9*x^2)*exp(x)+2*x^4-3*x^3) 
,x)
 

Output:

-64/(2*I*x^2*ln(x)-2*I*ln(2)+2*I*x+2*I*exp(x)-2*I*ln(x-3/2))^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (22) = 44\).

Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 3.26 \[ \int \frac {160+e^x (96-64 x)+32 x-64 x^2+\left (192 x-128 x^2\right ) \log (x)}{-3 x^3+2 x^4+e^{3 x} (-3+2 x)+e^{2 x} \left (-9 x+6 x^2\right )+e^x \left (-9 x^2+6 x^3\right )+\left (-3 x^6+2 x^7\right ) \log ^3(x)+\left (-9 x^2+6 x^3+e^{2 x} (-9+6 x)+e^x \left (-18 x+12 x^2\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x+6 x^2+e^x (-9+6 x)\right ) \log ^2\left (\frac {1}{-3+2 x}\right )+(-3+2 x) \log ^3\left (\frac {1}{-3+2 x}\right )+\log ^2(x) \left (-9 x^5+6 x^6+e^x \left (-9 x^4+6 x^5\right )+\left (-9 x^4+6 x^5\right ) \log \left (\frac {1}{-3+2 x}\right )\right )+\log (x) \left (-9 x^4+6 x^5+e^{2 x} \left (-9 x^2+6 x^3\right )+e^x \left (-18 x^3+12 x^4\right )+\left (-18 x^3+12 x^4+e^x \left (-18 x^2+12 x^3\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x^2+6 x^3\right ) \log ^2\left (\frac {1}{-3+2 x}\right )\right )} \, dx=\frac {16}{x^{4} \log \left (x\right )^{2} + x^{2} + 2 \, x e^{x} + 2 \, {\left (x^{3} + x^{2} e^{x} + x^{2} \log \left (\frac {1}{2 \, x - 3}\right )\right )} \log \left (x\right ) + 2 \, {\left (x + e^{x}\right )} \log \left (\frac {1}{2 \, x - 3}\right ) + \log \left (\frac {1}{2 \, x - 3}\right )^{2} + e^{\left (2 \, x\right )}} \] Input:

integrate(((-128*x^2+192*x)*log(x)+(-64*x+96)*exp(x)-64*x^2+32*x+160)/((2* 
x^7-3*x^6)*log(x)^3+((6*x^5-9*x^4)*log(1/(-3+2*x))+(6*x^5-9*x^4)*exp(x)+6* 
x^6-9*x^5)*log(x)^2+((6*x^3-9*x^2)*log(1/(-3+2*x))^2+((12*x^3-18*x^2)*exp( 
x)+12*x^4-18*x^3)*log(1/(-3+2*x))+(6*x^3-9*x^2)*exp(x)^2+(12*x^4-18*x^3)*e 
xp(x)+6*x^5-9*x^4)*log(x)+(-3+2*x)*log(1/(-3+2*x))^3+((6*x-9)*exp(x)+6*x^2 
-9*x)*log(1/(-3+2*x))^2+((6*x-9)*exp(x)^2+(12*x^2-18*x)*exp(x)+6*x^3-9*x^2 
)*log(1/(-3+2*x))+(-3+2*x)*exp(x)^3+(6*x^2-9*x)*exp(x)^2+(6*x^3-9*x^2)*exp 
(x)+2*x^4-3*x^3),x, algorithm="fricas")
 

Output:

16/(x^4*log(x)^2 + x^2 + 2*x*e^x + 2*(x^3 + x^2*e^x + x^2*log(1/(2*x - 3)) 
)*log(x) + 2*(x + e^x)*log(1/(2*x - 3)) + log(1/(2*x - 3))^2 + e^(2*x))
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (22) = 44\).

Time = 0.32 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.91 \[ \int \frac {160+e^x (96-64 x)+32 x-64 x^2+\left (192 x-128 x^2\right ) \log (x)}{-3 x^3+2 x^4+e^{3 x} (-3+2 x)+e^{2 x} \left (-9 x+6 x^2\right )+e^x \left (-9 x^2+6 x^3\right )+\left (-3 x^6+2 x^7\right ) \log ^3(x)+\left (-9 x^2+6 x^3+e^{2 x} (-9+6 x)+e^x \left (-18 x+12 x^2\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x+6 x^2+e^x (-9+6 x)\right ) \log ^2\left (\frac {1}{-3+2 x}\right )+(-3+2 x) \log ^3\left (\frac {1}{-3+2 x}\right )+\log ^2(x) \left (-9 x^5+6 x^6+e^x \left (-9 x^4+6 x^5\right )+\left (-9 x^4+6 x^5\right ) \log \left (\frac {1}{-3+2 x}\right )\right )+\log (x) \left (-9 x^4+6 x^5+e^{2 x} \left (-9 x^2+6 x^3\right )+e^x \left (-18 x^3+12 x^4\right )+\left (-18 x^3+12 x^4+e^x \left (-18 x^2+12 x^3\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x^2+6 x^3\right ) \log ^2\left (\frac {1}{-3+2 x}\right )\right )} \, dx=\frac {16}{x^{4} \log {\left (x \right )}^{2} + 2 x^{3} \log {\left (x \right )} + 2 x^{2} \log {\left (x \right )} \log {\left (\frac {1}{2 x - 3} \right )} + x^{2} + 2 x \log {\left (\frac {1}{2 x - 3} \right )} + \left (2 x^{2} \log {\left (x \right )} + 2 x + 2 \log {\left (\frac {1}{2 x - 3} \right )}\right ) e^{x} + e^{2 x} + \log {\left (\frac {1}{2 x - 3} \right )}^{2}} \] Input:

integrate(((-128*x**2+192*x)*ln(x)+(-64*x+96)*exp(x)-64*x**2+32*x+160)/((2 
*x**7-3*x**6)*ln(x)**3+((6*x**5-9*x**4)*ln(1/(-3+2*x))+(6*x**5-9*x**4)*exp 
(x)+6*x**6-9*x**5)*ln(x)**2+((6*x**3-9*x**2)*ln(1/(-3+2*x))**2+((12*x**3-1 
8*x**2)*exp(x)+12*x**4-18*x**3)*ln(1/(-3+2*x))+(6*x**3-9*x**2)*exp(x)**2+( 
12*x**4-18*x**3)*exp(x)+6*x**5-9*x**4)*ln(x)+(-3+2*x)*ln(1/(-3+2*x))**3+(( 
6*x-9)*exp(x)+6*x**2-9*x)*ln(1/(-3+2*x))**2+((6*x-9)*exp(x)**2+(12*x**2-18 
*x)*exp(x)+6*x**3-9*x**2)*ln(1/(-3+2*x))+(-3+2*x)*exp(x)**3+(6*x**2-9*x)*e 
xp(x)**2+(6*x**3-9*x**2)*exp(x)+2*x**4-3*x**3),x)
 

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (22) = 44\).

Time = 0.40 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.83 \[ \int \frac {160+e^x (96-64 x)+32 x-64 x^2+\left (192 x-128 x^2\right ) \log (x)}{-3 x^3+2 x^4+e^{3 x} (-3+2 x)+e^{2 x} \left (-9 x+6 x^2\right )+e^x \left (-9 x^2+6 x^3\right )+\left (-3 x^6+2 x^7\right ) \log ^3(x)+\left (-9 x^2+6 x^3+e^{2 x} (-9+6 x)+e^x \left (-18 x+12 x^2\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x+6 x^2+e^x (-9+6 x)\right ) \log ^2\left (\frac {1}{-3+2 x}\right )+(-3+2 x) \log ^3\left (\frac {1}{-3+2 x}\right )+\log ^2(x) \left (-9 x^5+6 x^6+e^x \left (-9 x^4+6 x^5\right )+\left (-9 x^4+6 x^5\right ) \log \left (\frac {1}{-3+2 x}\right )\right )+\log (x) \left (-9 x^4+6 x^5+e^{2 x} \left (-9 x^2+6 x^3\right )+e^x \left (-18 x^3+12 x^4\right )+\left (-18 x^3+12 x^4+e^x \left (-18 x^2+12 x^3\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x^2+6 x^3\right ) \log ^2\left (\frac {1}{-3+2 x}\right )\right )} \, dx=\frac {16}{x^{4} \log \left (x\right )^{2} + 2 \, x^{3} \log \left (x\right ) + x^{2} + 2 \, {\left (x^{2} \log \left (x\right ) + x\right )} e^{x} - 2 \, {\left (x^{2} \log \left (x\right ) + x + e^{x}\right )} \log \left (2 \, x - 3\right ) + \log \left (2 \, x - 3\right )^{2} + e^{\left (2 \, x\right )}} \] Input:

integrate(((-128*x^2+192*x)*log(x)+(-64*x+96)*exp(x)-64*x^2+32*x+160)/((2* 
x^7-3*x^6)*log(x)^3+((6*x^5-9*x^4)*log(1/(-3+2*x))+(6*x^5-9*x^4)*exp(x)+6* 
x^6-9*x^5)*log(x)^2+((6*x^3-9*x^2)*log(1/(-3+2*x))^2+((12*x^3-18*x^2)*exp( 
x)+12*x^4-18*x^3)*log(1/(-3+2*x))+(6*x^3-9*x^2)*exp(x)^2+(12*x^4-18*x^3)*e 
xp(x)+6*x^5-9*x^4)*log(x)+(-3+2*x)*log(1/(-3+2*x))^3+((6*x-9)*exp(x)+6*x^2 
-9*x)*log(1/(-3+2*x))^2+((6*x-9)*exp(x)^2+(12*x^2-18*x)*exp(x)+6*x^3-9*x^2 
)*log(1/(-3+2*x))+(-3+2*x)*exp(x)^3+(6*x^2-9*x)*exp(x)^2+(6*x^3-9*x^2)*exp 
(x)+2*x^4-3*x^3),x, algorithm="maxima")
 

Output:

16/(x^4*log(x)^2 + 2*x^3*log(x) + x^2 + 2*(x^2*log(x) + x)*e^x - 2*(x^2*lo 
g(x) + x + e^x)*log(2*x - 3) + log(2*x - 3)^2 + e^(2*x))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 81 vs. \(2 (22) = 44\).

Time = 1.08 (sec) , antiderivative size = 81, normalized size of antiderivative = 3.52 \[ \int \frac {160+e^x (96-64 x)+32 x-64 x^2+\left (192 x-128 x^2\right ) \log (x)}{-3 x^3+2 x^4+e^{3 x} (-3+2 x)+e^{2 x} \left (-9 x+6 x^2\right )+e^x \left (-9 x^2+6 x^3\right )+\left (-3 x^6+2 x^7\right ) \log ^3(x)+\left (-9 x^2+6 x^3+e^{2 x} (-9+6 x)+e^x \left (-18 x+12 x^2\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x+6 x^2+e^x (-9+6 x)\right ) \log ^2\left (\frac {1}{-3+2 x}\right )+(-3+2 x) \log ^3\left (\frac {1}{-3+2 x}\right )+\log ^2(x) \left (-9 x^5+6 x^6+e^x \left (-9 x^4+6 x^5\right )+\left (-9 x^4+6 x^5\right ) \log \left (\frac {1}{-3+2 x}\right )\right )+\log (x) \left (-9 x^4+6 x^5+e^{2 x} \left (-9 x^2+6 x^3\right )+e^x \left (-18 x^3+12 x^4\right )+\left (-18 x^3+12 x^4+e^x \left (-18 x^2+12 x^3\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x^2+6 x^3\right ) \log ^2\left (\frac {1}{-3+2 x}\right )\right )} \, dx=\frac {16}{x^{4} \log \left (x\right )^{2} + 2 \, x^{3} \log \left (x\right ) + 2 \, x^{2} e^{x} \log \left (x\right ) - 2 \, x^{2} \log \left (2 \, x - 3\right ) \log \left (x\right ) + x^{2} + 2 \, x e^{x} - 2 \, x \log \left (2 \, x - 3\right ) - 2 \, e^{x} \log \left (2 \, x - 3\right ) + \log \left (2 \, x - 3\right )^{2} + e^{\left (2 \, x\right )}} \] Input:

integrate(((-128*x^2+192*x)*log(x)+(-64*x+96)*exp(x)-64*x^2+32*x+160)/((2* 
x^7-3*x^6)*log(x)^3+((6*x^5-9*x^4)*log(1/(-3+2*x))+(6*x^5-9*x^4)*exp(x)+6* 
x^6-9*x^5)*log(x)^2+((6*x^3-9*x^2)*log(1/(-3+2*x))^2+((12*x^3-18*x^2)*exp( 
x)+12*x^4-18*x^3)*log(1/(-3+2*x))+(6*x^3-9*x^2)*exp(x)^2+(12*x^4-18*x^3)*e 
xp(x)+6*x^5-9*x^4)*log(x)+(-3+2*x)*log(1/(-3+2*x))^3+((6*x-9)*exp(x)+6*x^2 
-9*x)*log(1/(-3+2*x))^2+((6*x-9)*exp(x)^2+(12*x^2-18*x)*exp(x)+6*x^3-9*x^2 
)*log(1/(-3+2*x))+(-3+2*x)*exp(x)^3+(6*x^2-9*x)*exp(x)^2+(6*x^3-9*x^2)*exp 
(x)+2*x^4-3*x^3),x, algorithm="giac")
 

Output:

16/(x^4*log(x)^2 + 2*x^3*log(x) + 2*x^2*e^x*log(x) - 2*x^2*log(2*x - 3)*lo 
g(x) + x^2 + 2*x*e^x - 2*x*log(2*x - 3) - 2*e^x*log(2*x - 3) + log(2*x - 3 
)^2 + e^(2*x))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {160+e^x (96-64 x)+32 x-64 x^2+\left (192 x-128 x^2\right ) \log (x)}{-3 x^3+2 x^4+e^{3 x} (-3+2 x)+e^{2 x} \left (-9 x+6 x^2\right )+e^x \left (-9 x^2+6 x^3\right )+\left (-3 x^6+2 x^7\right ) \log ^3(x)+\left (-9 x^2+6 x^3+e^{2 x} (-9+6 x)+e^x \left (-18 x+12 x^2\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x+6 x^2+e^x (-9+6 x)\right ) \log ^2\left (\frac {1}{-3+2 x}\right )+(-3+2 x) \log ^3\left (\frac {1}{-3+2 x}\right )+\log ^2(x) \left (-9 x^5+6 x^6+e^x \left (-9 x^4+6 x^5\right )+\left (-9 x^4+6 x^5\right ) \log \left (\frac {1}{-3+2 x}\right )\right )+\log (x) \left (-9 x^4+6 x^5+e^{2 x} \left (-9 x^2+6 x^3\right )+e^x \left (-18 x^3+12 x^4\right )+\left (-18 x^3+12 x^4+e^x \left (-18 x^2+12 x^3\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x^2+6 x^3\right ) \log ^2\left (\frac {1}{-3+2 x}\right )\right )} \, dx=\int -\frac {32\,x-{\mathrm {e}}^x\,\left (64\,x-96\right )+\ln \left (x\right )\,\left (192\,x-128\,x^2\right )-64\,x^2+160}{{\mathrm {e}}^{2\,x}\,\left (9\,x-6\,x^2\right )+{\mathrm {e}}^x\,\left (9\,x^2-6\,x^3\right )+\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (18\,x^3-12\,x^4\right )+\ln \left (\frac {1}{2\,x-3}\right )\,\left ({\mathrm {e}}^x\,\left (18\,x^2-12\,x^3\right )+18\,x^3-12\,x^4\right )+{\mathrm {e}}^{2\,x}\,\left (9\,x^2-6\,x^3\right )+{\ln \left (\frac {1}{2\,x-3}\right )}^2\,\left (9\,x^2-6\,x^3\right )+9\,x^4-6\,x^5\right )+{\ln \left (x\right )}^3\,\left (3\,x^6-2\,x^7\right )-{\ln \left (\frac {1}{2\,x-3}\right )}^2\,\left ({\mathrm {e}}^x\,\left (6\,x-9\right )-9\,x+6\,x^2\right )+{\ln \left (x\right )}^2\,\left ({\mathrm {e}}^x\,\left (9\,x^4-6\,x^5\right )+\ln \left (\frac {1}{2\,x-3}\right )\,\left (9\,x^4-6\,x^5\right )+9\,x^5-6\,x^6\right )+\ln \left (\frac {1}{2\,x-3}\right )\,\left ({\mathrm {e}}^x\,\left (18\,x-12\,x^2\right )-{\mathrm {e}}^{2\,x}\,\left (6\,x-9\right )+9\,x^2-6\,x^3\right )-{\mathrm {e}}^{3\,x}\,\left (2\,x-3\right )+3\,x^3-2\,x^4-{\ln \left (\frac {1}{2\,x-3}\right )}^3\,\left (2\,x-3\right )} \,d x \] Input:

int(-(32*x - exp(x)*(64*x - 96) + log(x)*(192*x - 128*x^2) - 64*x^2 + 160) 
/(exp(2*x)*(9*x - 6*x^2) + exp(x)*(9*x^2 - 6*x^3) + log(x)*(exp(x)*(18*x^3 
 - 12*x^4) + log(1/(2*x - 3))*(exp(x)*(18*x^2 - 12*x^3) + 18*x^3 - 12*x^4) 
 + exp(2*x)*(9*x^2 - 6*x^3) + log(1/(2*x - 3))^2*(9*x^2 - 6*x^3) + 9*x^4 - 
 6*x^5) + log(x)^3*(3*x^6 - 2*x^7) - log(1/(2*x - 3))^2*(exp(x)*(6*x - 9) 
- 9*x + 6*x^2) + log(x)^2*(exp(x)*(9*x^4 - 6*x^5) + log(1/(2*x - 3))*(9*x^ 
4 - 6*x^5) + 9*x^5 - 6*x^6) + log(1/(2*x - 3))*(exp(x)*(18*x - 12*x^2) - e 
xp(2*x)*(6*x - 9) + 9*x^2 - 6*x^3) - exp(3*x)*(2*x - 3) + 3*x^3 - 2*x^4 - 
log(1/(2*x - 3))^3*(2*x - 3)),x)
 

Output:

int(-(32*x - exp(x)*(64*x - 96) + log(x)*(192*x - 128*x^2) - 64*x^2 + 160) 
/(exp(2*x)*(9*x - 6*x^2) + exp(x)*(9*x^2 - 6*x^3) + log(x)*(exp(x)*(18*x^3 
 - 12*x^4) + log(1/(2*x - 3))*(exp(x)*(18*x^2 - 12*x^3) + 18*x^3 - 12*x^4) 
 + exp(2*x)*(9*x^2 - 6*x^3) + log(1/(2*x - 3))^2*(9*x^2 - 6*x^3) + 9*x^4 - 
 6*x^5) + log(x)^3*(3*x^6 - 2*x^7) - log(1/(2*x - 3))^2*(exp(x)*(6*x - 9) 
- 9*x + 6*x^2) + log(x)^2*(exp(x)*(9*x^4 - 6*x^5) + log(1/(2*x - 3))*(9*x^ 
4 - 6*x^5) + 9*x^5 - 6*x^6) + log(1/(2*x - 3))*(exp(x)*(18*x - 12*x^2) - e 
xp(2*x)*(6*x - 9) + 9*x^2 - 6*x^3) - exp(3*x)*(2*x - 3) + 3*x^3 - 2*x^4 - 
log(1/(2*x - 3))^3*(2*x - 3)), x)
 

Reduce [B] (verification not implemented)

Time = 0.43 (sec) , antiderivative size = 85, normalized size of antiderivative = 3.70 \[ \int \frac {160+e^x (96-64 x)+32 x-64 x^2+\left (192 x-128 x^2\right ) \log (x)}{-3 x^3+2 x^4+e^{3 x} (-3+2 x)+e^{2 x} \left (-9 x+6 x^2\right )+e^x \left (-9 x^2+6 x^3\right )+\left (-3 x^6+2 x^7\right ) \log ^3(x)+\left (-9 x^2+6 x^3+e^{2 x} (-9+6 x)+e^x \left (-18 x+12 x^2\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x+6 x^2+e^x (-9+6 x)\right ) \log ^2\left (\frac {1}{-3+2 x}\right )+(-3+2 x) \log ^3\left (\frac {1}{-3+2 x}\right )+\log ^2(x) \left (-9 x^5+6 x^6+e^x \left (-9 x^4+6 x^5\right )+\left (-9 x^4+6 x^5\right ) \log \left (\frac {1}{-3+2 x}\right )\right )+\log (x) \left (-9 x^4+6 x^5+e^{2 x} \left (-9 x^2+6 x^3\right )+e^x \left (-18 x^3+12 x^4\right )+\left (-18 x^3+12 x^4+e^x \left (-18 x^2+12 x^3\right )\right ) \log \left (\frac {1}{-3+2 x}\right )+\left (-9 x^2+6 x^3\right ) \log ^2\left (\frac {1}{-3+2 x}\right )\right )} \, dx=\frac {16}{e^{2 x}-2 e^{x} \mathrm {log}\left (2 x -3\right )+2 e^{x} \mathrm {log}\left (x \right ) x^{2}+2 e^{x} x +\mathrm {log}\left (2 x -3\right )^{2}-2 \,\mathrm {log}\left (2 x -3\right ) \mathrm {log}\left (x \right ) x^{2}-2 \,\mathrm {log}\left (2 x -3\right ) x +\mathrm {log}\left (x \right )^{2} x^{4}+2 \,\mathrm {log}\left (x \right ) x^{3}+x^{2}} \] Input:

int(((-128*x^2+192*x)*log(x)+(-64*x+96)*exp(x)-64*x^2+32*x+160)/((2*x^7-3* 
x^6)*log(x)^3+((6*x^5-9*x^4)*log(1/(-3+2*x))+(6*x^5-9*x^4)*exp(x)+6*x^6-9* 
x^5)*log(x)^2+((6*x^3-9*x^2)*log(1/(-3+2*x))^2+((12*x^3-18*x^2)*exp(x)+12* 
x^4-18*x^3)*log(1/(-3+2*x))+(6*x^3-9*x^2)*exp(x)^2+(12*x^4-18*x^3)*exp(x)+ 
6*x^5-9*x^4)*log(x)+(-3+2*x)*log(1/(-3+2*x))^3+((6*x-9)*exp(x)+6*x^2-9*x)* 
log(1/(-3+2*x))^2+((6*x-9)*exp(x)^2+(12*x^2-18*x)*exp(x)+6*x^3-9*x^2)*log( 
1/(-3+2*x))+(-3+2*x)*exp(x)^3+(6*x^2-9*x)*exp(x)^2+(6*x^3-9*x^2)*exp(x)+2* 
x^4-3*x^3),x)
 

Output:

16/(e**(2*x) - 2*e**x*log(2*x - 3) + 2*e**x*log(x)*x**2 + 2*e**x*x + log(2 
*x - 3)**2 - 2*log(2*x - 3)*log(x)*x**2 - 2*log(2*x - 3)*x + log(x)**2*x** 
4 + 2*log(x)*x**3 + x**2)