\(\int \frac {e^3 (-100 x^3+24 x^4)-2 e^3 x^4 \log (4)+2 e^3 x^4 \log (4) \log (x)}{-15625+22500 x-10800 x^2+1728 x^3+(1875 x-1800 x^2+432 x^3) \log (4) \log (x)+(-75 x^2+36 x^3) \log ^2(4) \log ^2(x)+x^3 \log ^3(4) \log ^3(x)} \, dx\) [759]

Optimal result

Integrand size = 107, antiderivative size = 29 \[ \int \frac {e^3 \left (-100 x^3+24 x^4\right )-2 e^3 x^4 \log (4)+2 e^3 x^4 \log (4) \log (x)}{-15625+22500 x-10800 x^2+1728 x^3+\left (1875 x-1800 x^2+432 x^3\right ) \log (4) \log (x)+\left (-75 x^2+36 x^3\right ) \log ^2(4) \log ^2(x)+x^3 \log ^3(4) \log ^3(x)} \, dx=\frac {e^3 x^2}{\left (-2+\frac {(-5+x)^2}{x}-x-\log (4) \log (x)\right )^2} \] Output:

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

Mathematica [A] (verified)

Time = 5.01 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.69 \[ \int \frac {e^3 \left (-100 x^3+24 x^4\right )-2 e^3 x^4 \log (4)+2 e^3 x^4 \log (4) \log (x)}{-15625+22500 x-10800 x^2+1728 x^3+\left (1875 x-1800 x^2+432 x^3\right ) \log (4) \log (x)+\left (-75 x^2+36 x^3\right ) \log ^2(4) \log ^2(x)+x^3 \log ^3(4) \log ^3(x)} \, dx=\frac {e^3 x^4}{(-25+12 x+x \log (4) \log (x))^2} \] Input:

Integrate[(E^3*(-100*x^3 + 24*x^4) - 2*E^3*x^4*Log[4] + 2*E^3*x^4*Log[4]*L 
og[x])/(-15625 + 22500*x - 10800*x^2 + 1728*x^3 + (1875*x - 1800*x^2 + 432 
*x^3)*Log[4]*Log[x] + (-75*x^2 + 36*x^3)*Log[4]^2*Log[x]^2 + x^3*Log[4]^3* 
Log[x]^3),x]
 

Output:

(E^3*x^4)/(-25 + 12*x + x*Log[4]*Log[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.

\(\displaystyle \int \frac {2 e^3 x^4 \log (4) \log (x)-2 e^3 x^4 \log (4)+e^3 \left (24 x^4-100 x^3\right )}{1728 x^3+x^3 \log ^3(4) \log ^3(x)-10800 x^2+\left (36 x^3-75 x^2\right ) \log ^2(4) \log ^2(x)+\left (432 x^3-1800 x^2+1875 x\right ) \log (4) \log (x)+22500 x-15625} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 e^3 x^3 (x (-\log (4)) \log (x)+x (\log (4)-12)+50)}{(-12 x+x (-\log (4)) \log (x)+25)^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 e^3 \int \frac {x^3 (-\log (4) \log (x) x-(12-\log (4)) x+50)}{(-\log (4) \log (x) x-12 x+25)^3}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle 2 e^3 \int \left (\frac {x^3}{(\log (4) \log (x) x+12 x-25)^2}-\frac {x^3 (\log (4) x+25)}{(\log (4) \log (x) x+12 x-25)^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 e^3 \left (-\log (4) \int \frac {x^4}{(\log (4) \log (x) x+12 x-25)^3}dx-25 \int \frac {x^3}{(\log (4) \log (x) x+12 x-25)^3}dx+\int \frac {x^3}{(\log (4) \log (x) x+12 x-25)^2}dx\right )\)

Input:

Int[(E^3*(-100*x^3 + 24*x^4) - 2*E^3*x^4*Log[4] + 2*E^3*x^4*Log[4]*Log[x]) 
/(-15625 + 22500*x - 10800*x^2 + 1728*x^3 + (1875*x - 1800*x^2 + 432*x^3)* 
Log[4]*Log[x] + (-75*x^2 + 36*x^3)*Log[4]^2*Log[x]^2 + x^3*Log[4]^3*Log[x] 
^3),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.87 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72

method	result	size
default	\(\frac {{\mathrm e}^{3} x^{4}}{\left (2 x \ln \left (2\right ) \ln \left (x \right )+12 x -25\right )^{2}}\)	\(21\)
norman	\(\frac {{\mathrm e}^{3} x^{4}}{\left (2 x \ln \left (2\right ) \ln \left (x \right )+12 x -25\right )^{2}}\)	\(21\)
risch	\(\frac {{\mathrm e}^{3} x^{4}}{\left (2 x \ln \left (2\right ) \ln \left (x \right )+12 x -25\right )^{2}}\)	\(21\)
parallelrisch	\(\frac {x^{4} {\mathrm e}^{3}}{4 x^{2} \ln \left (2\right )^{2} \ln \left (x \right )^{2}+48 x^{2} \ln \left (2\right ) \ln \left (x \right )-100 x \ln \left (2\right ) \ln \left (x \right )+144 x^{2}-600 x +625}\)	\(48\)

Input:

int((4*x^4*exp(3)*ln(2)*ln(x)-4*x^4*exp(3)*ln(2)+(24*x^4-100*x^3)*exp(3))/ 
(8*x^3*ln(2)^3*ln(x)^3+4*(36*x^3-75*x^2)*ln(2)^2*ln(x)^2+2*(432*x^3-1800*x 
^2+1875*x)*ln(2)*ln(x)+1728*x^3-10800*x^2+22500*x-15625),x,method=_RETURNV 
ERBOSE)
 

Output:

exp(3)*x^4/(2*x*ln(2)*ln(x)+12*x-25)^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {e^3 \left (-100 x^3+24 x^4\right )-2 e^3 x^4 \log (4)+2 e^3 x^4 \log (4) \log (x)}{-15625+22500 x-10800 x^2+1728 x^3+\left (1875 x-1800 x^2+432 x^3\right ) \log (4) \log (x)+\left (-75 x^2+36 x^3\right ) \log ^2(4) \log ^2(x)+x^3 \log ^3(4) \log ^3(x)} \, dx=\frac {x^{4} e^{3}}{4 \, x^{2} \log \left (2\right )^{2} \log \left (x\right )^{2} + 4 \, {\left (12 \, x^{2} - 25 \, x\right )} \log \left (2\right ) \log \left (x\right ) + 144 \, x^{2} - 600 \, x + 625} \] Input:

integrate((4*x^4*exp(3)*log(2)*log(x)-4*x^4*exp(3)*log(2)+(24*x^4-100*x^3) 
*exp(3))/(8*x^3*log(2)^3*log(x)^3+4*(36*x^3-75*x^2)*log(2)^2*log(x)^2+2*(4 
32*x^3-1800*x^2+1875*x)*log(2)*log(x)+1728*x^3-10800*x^2+22500*x-15625),x, 
 algorithm="fricas")
 

Output:

x^4*e^3/(4*x^2*log(2)^2*log(x)^2 + 4*(12*x^2 - 25*x)*log(2)*log(x) + 144*x 
^2 - 600*x + 625)
 

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.69 \[ \int \frac {e^3 \left (-100 x^3+24 x^4\right )-2 e^3 x^4 \log (4)+2 e^3 x^4 \log (4) \log (x)}{-15625+22500 x-10800 x^2+1728 x^3+\left (1875 x-1800 x^2+432 x^3\right ) \log (4) \log (x)+\left (-75 x^2+36 x^3\right ) \log ^2(4) \log ^2(x)+x^3 \log ^3(4) \log ^3(x)} \, dx=\frac {x^{4} e^{3}}{4 x^{2} \log {\left (2 \right )}^{2} \log {\left (x \right )}^{2} + 144 x^{2} - 600 x + \left (48 x^{2} \log {\left (2 \right )} - 100 x \log {\left (2 \right )}\right ) \log {\left (x \right )} + 625} \] Input:

integrate((4*x**4*exp(3)*ln(2)*ln(x)-4*x**4*exp(3)*ln(2)+(24*x**4-100*x**3 
)*exp(3))/(8*x**3*ln(2)**3*ln(x)**3+4*(36*x**3-75*x**2)*ln(2)**2*ln(x)**2+ 
2*(432*x**3-1800*x**2+1875*x)*ln(2)*ln(x)+1728*x**3-10800*x**2+22500*x-156 
25),x)
 

Output:

x**4*exp(3)/(4*x**2*log(2)**2*log(x)**2 + 144*x**2 - 600*x + (48*x**2*log( 
2) - 100*x*log(2))*log(x) + 625)
 

Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {e^3 \left (-100 x^3+24 x^4\right )-2 e^3 x^4 \log (4)+2 e^3 x^4 \log (4) \log (x)}{-15625+22500 x-10800 x^2+1728 x^3+\left (1875 x-1800 x^2+432 x^3\right ) \log (4) \log (x)+\left (-75 x^2+36 x^3\right ) \log ^2(4) \log ^2(x)+x^3 \log ^3(4) \log ^3(x)} \, dx=\frac {x^{4} e^{3}}{4 \, x^{2} \log \left (2\right )^{2} \log \left (x\right )^{2} + 144 \, x^{2} + 4 \, {\left (12 \, x^{2} \log \left (2\right ) - 25 \, x \log \left (2\right )\right )} \log \left (x\right ) - 600 \, x + 625} \] Input:

integrate((4*x^4*exp(3)*log(2)*log(x)-4*x^4*exp(3)*log(2)+(24*x^4-100*x^3) 
*exp(3))/(8*x^3*log(2)^3*log(x)^3+4*(36*x^3-75*x^2)*log(2)^2*log(x)^2+2*(4 
32*x^3-1800*x^2+1875*x)*log(2)*log(x)+1728*x^3-10800*x^2+22500*x-15625),x, 
 algorithm="maxima")
 

Output:

x^4*e^3/(4*x^2*log(2)^2*log(x)^2 + 144*x^2 + 4*(12*x^2*log(2) - 25*x*log(2 
))*log(x) - 600*x + 625)
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {e^3 \left (-100 x^3+24 x^4\right )-2 e^3 x^4 \log (4)+2 e^3 x^4 \log (4) \log (x)}{-15625+22500 x-10800 x^2+1728 x^3+\left (1875 x-1800 x^2+432 x^3\right ) \log (4) \log (x)+\left (-75 x^2+36 x^3\right ) \log ^2(4) \log ^2(x)+x^3 \log ^3(4) \log ^3(x)} \, dx=\frac {x^{4} e^{3}}{4 \, x^{2} \log \left (2\right )^{2} \log \left (x\right )^{2} + 48 \, x^{2} \log \left (2\right ) \log \left (x\right ) - 100 \, x \log \left (2\right ) \log \left (x\right ) + 144 \, x^{2} - 600 \, x + 625} \] Input:

integrate((4*x^4*exp(3)*log(2)*log(x)-4*x^4*exp(3)*log(2)+(24*x^4-100*x^3) 
*exp(3))/(8*x^3*log(2)^3*log(x)^3+4*(36*x^3-75*x^2)*log(2)^2*log(x)^2+2*(4 
32*x^3-1800*x^2+1875*x)*log(2)*log(x)+1728*x^3-10800*x^2+22500*x-15625),x, 
 algorithm="giac")
 

Output:

x^4*e^3/(4*x^2*log(2)^2*log(x)^2 + 48*x^2*log(2)*log(x) - 100*x*log(2)*log 
(x) + 144*x^2 - 600*x + 625)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {e^3 \left (-100 x^3+24 x^4\right )-2 e^3 x^4 \log (4)+2 e^3 x^4 \log (4) \log (x)}{-15625+22500 x-10800 x^2+1728 x^3+\left (1875 x-1800 x^2+432 x^3\right ) \log (4) \log (x)+\left (-75 x^2+36 x^3\right ) \log ^2(4) \log ^2(x)+x^3 \log ^3(4) \log ^3(x)} \, dx=\int -\frac {{\mathrm {e}}^3\,\left (100\,x^3-24\,x^4\right )+4\,x^4\,{\mathrm {e}}^3\,\ln \left (2\right )-4\,x^4\,{\mathrm {e}}^3\,\ln \left (2\right )\,\ln \left (x\right )}{22500\,x-10800\,x^2+1728\,x^3-4\,{\ln \left (2\right )}^2\,{\ln \left (x\right )}^2\,\left (75\,x^2-36\,x^3\right )+8\,x^3\,{\ln \left (2\right )}^3\,{\ln \left (x\right )}^3+2\,\ln \left (2\right )\,\ln \left (x\right )\,\left (432\,x^3-1800\,x^2+1875\,x\right )-15625} \,d x \] Input:

int(-(exp(3)*(100*x^3 - 24*x^4) + 4*x^4*exp(3)*log(2) - 4*x^4*exp(3)*log(2 
)*log(x))/(22500*x - 10800*x^2 + 1728*x^3 - 4*log(2)^2*log(x)^2*(75*x^2 - 
36*x^3) + 8*x^3*log(2)^3*log(x)^3 + 2*log(2)*log(x)*(1875*x - 1800*x^2 + 4 
32*x^3) - 15625),x)
 

Output:

int(-(exp(3)*(100*x^3 - 24*x^4) + 4*x^4*exp(3)*log(2) - 4*x^4*exp(3)*log(2 
)*log(x))/(22500*x - 10800*x^2 + 1728*x^3 - 4*log(2)^2*log(x)^2*(75*x^2 - 
36*x^3) + 8*x^3*log(2)^3*log(x)^3 + 2*log(2)*log(x)*(1875*x - 1800*x^2 + 4 
32*x^3) - 15625), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {e^3 \left (-100 x^3+24 x^4\right )-2 e^3 x^4 \log (4)+2 e^3 x^4 \log (4) \log (x)}{-15625+22500 x-10800 x^2+1728 x^3+\left (1875 x-1800 x^2+432 x^3\right ) \log (4) \log (x)+\left (-75 x^2+36 x^3\right ) \log ^2(4) \log ^2(x)+x^3 \log ^3(4) \log ^3(x)} \, dx=\frac {e^{3} x^{4}}{4 \mathrm {log}\left (x \right )^{2} \mathrm {log}\left (2\right )^{2} x^{2}+48 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x^{2}-100 \,\mathrm {log}\left (x \right ) \mathrm {log}\left (2\right ) x +144 x^{2}-600 x +625} \] Input:

int((4*x^4*exp(3)*log(2)*log(x)-4*x^4*exp(3)*log(2)+(24*x^4-100*x^3)*exp(3 
))/(8*x^3*log(2)^3*log(x)^3+4*(36*x^3-75*x^2)*log(2)^2*log(x)^2+2*(432*x^3 
-1800*x^2+1875*x)*log(2)*log(x)+1728*x^3-10800*x^2+22500*x-15625),x)
 

Output:

(e**3*x**4)/(4*log(x)**2*log(2)**2*x**2 + 48*log(x)*log(2)*x**2 - 100*log( 
x)*log(2)*x + 144*x**2 - 600*x + 625)