\(\int \frac {e^6 x^2 (-40+100 x-102 x^2+48 x^3-10 x^4+(40-50 x+34 x^2-12 x^3+2 x^4) \log (20 x-25 x^2+17 x^3-6 x^4+x^5))}{(20 x-25 x^2+17 x^3-6 x^4+x^5) \log ^3(20 x-25 x^2+17 x^3-6 x^4+x^5)} \, dx\) [2947]

Optimal result

Integrand size = 120, antiderivative size = 26 \[ \int \frac {e^6 x^2 \left (-40+100 x-102 x^2+48 x^3-10 x^4+\left (40-50 x+34 x^2-12 x^3+2 x^4\right ) \log \left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )\right )}{\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right ) \log ^3\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )} \, dx=\frac {e^6 x^2}{\log ^2\left (x \left (4-x+\left ((-2+x)^2+x\right )^2\right )\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.15 \[ \int \frac {e^6 x^2 \left (-40+100 x-102 x^2+48 x^3-10 x^4+\left (40-50 x+34 x^2-12 x^3+2 x^4\right ) \log \left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )\right )}{\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right ) \log ^3\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )} \, dx=\frac {e^6 x^2}{\log ^2\left (x \left (20-25 x+17 x^2-6 x^3+x^4\right )\right )} \] Input:

Integrate[(E^6*x^2*(-40 + 100*x - 102*x^2 + 48*x^3 - 10*x^4 + (40 - 50*x + 
 34*x^2 - 12*x^3 + 2*x^4)*Log[20*x - 25*x^2 + 17*x^3 - 6*x^4 + x^5]))/((20 
*x - 25*x^2 + 17*x^3 - 6*x^4 + x^5)*Log[20*x - 25*x^2 + 17*x^3 - 6*x^4 + x 
^5]^3),x]
 

Output:

(E^6*x^2)/Log[x*(20 - 25*x + 17*x^2 - 6*x^3 + x^4)]^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.

\(\displaystyle \int \frac {e^6 x^2 \left (-10 x^4+48 x^3-102 x^2+\left (2 x^4-12 x^3+34 x^2-50 x+40\right ) \log \left (x^5-6 x^4+17 x^3-25 x^2+20 x\right )+100 x-40\right )}{\left (x^5-6 x^4+17 x^3-25 x^2+20 x\right ) \log ^3\left (x^5-6 x^4+17 x^3-25 x^2+20 x\right )} \, dx\)

\(\Big \downarrow \) 9

\(\displaystyle \int -\frac {2 e^6 x \left (5 x^4-24 x^3+51 x^2-\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log \left (x^5-6 x^4+17 x^3-25 x^2+20 x\right )-50 x+20\right )}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x^5-6 x^4+17 x^3-25 x^2+20 x\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle -2 e^6 \int \frac {x \left (5 x^4-24 x^3+51 x^2-50 x-\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log \left (x^5-6 x^4+17 x^3-25 x^2+20 x\right )+20\right )}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x^5-6 x^4+17 x^3-25 x^2+20 x\right )}dx\)

\(\Big \downarrow \) 7292

\(\displaystyle -2 e^6 \int \frac {x \left (5 x^4-24 x^3+51 x^2-50 x-\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log \left (x^5-6 x^4+17 x^3-25 x^2+20 x\right )+20\right )}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 e^6 \int \left (\frac {5 x^5}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}-\frac {24 x^4}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}+\frac {51 x^3}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}-\frac {50 x^2}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}-\frac {x}{\log ^2\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}+\frac {20 x}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 e^6 \left (6 \int \frac {1}{\log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}dx+5 \int \frac {x}{\log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}dx-120 \int \frac {1}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}dx+70 \int \frac {x}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}dx-27 \int \frac {x^2}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}dx+2 \int \frac {x^3}{\left (x^4-6 x^3+17 x^2-25 x+20\right ) \log ^3\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}dx-\int \frac {x}{\log ^2\left (x \left (x^4-6 x^3+17 x^2-25 x+20\right )\right )}dx\right )\)

Input:

Int[(E^6*x^2*(-40 + 100*x - 102*x^2 + 48*x^3 - 10*x^4 + (40 - 50*x + 34*x^ 
2 - 12*x^3 + 2*x^4)*Log[20*x - 25*x^2 + 17*x^3 - 6*x^4 + x^5]))/((20*x - 2 
5*x^2 + 17*x^3 - 6*x^4 + x^5)*Log[20*x - 25*x^2 + 17*x^3 - 6*x^4 + x^5]^3) 
,x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 22.61 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.31

method	result	size
parallelrisch	\(\frac {x^{2} {\mathrm e}^{6}}{\ln \left (x^{5}-6 x^{4}+17 x^{3}-25 x^{2}+20 x \right )^{2}}\)	\(34\)
risch	\(-\frac {4 \,{\mathrm e}^{6} x^{2}}{{\left (\pi \,\operatorname {csgn}\left (i \left (x^{4}-6 x^{3}+17 x^{2}-25 x +20\right )\right ) {\operatorname {csgn}\left (i x \left (x^{4}-6 x^{3}+17 x^{2}-25 x +20\right )\right )}^{2}-\pi \,\operatorname {csgn}\left (i \left (x^{4}-6 x^{3}+17 x^{2}-25 x +20\right )\right ) \operatorname {csgn}\left (i x \left (x^{4}-6 x^{3}+17 x^{2}-25 x +20\right )\right ) \operatorname {csgn}\left (i x \right )-\pi {\operatorname {csgn}\left (i x \left (x^{4}-6 x^{3}+17 x^{2}-25 x +20\right )\right )}^{3}+\pi {\operatorname {csgn}\left (i x \left (x^{4}-6 x^{3}+17 x^{2}-25 x +20\right )\right )}^{2} \operatorname {csgn}\left (i x \right )-2 i \ln \left (x \right )-2 i \ln \left (x^{4}-6 x^{3}+17 x^{2}-25 x +20\right )\right )}^{2}}\)	\(200\)

Input:

int(((2*x^4-12*x^3+34*x^2-50*x+40)*ln(x^5-6*x^4+17*x^3-25*x^2+20*x)-10*x^4 
+48*x^3-102*x^2+100*x-40)*exp(3+ln(x))^2/(x^5-6*x^4+17*x^3-25*x^2+20*x)/ln 
(x^5-6*x^4+17*x^3-25*x^2+20*x)^3,x,method=_RETURNVERBOSE)
 

Output:

exp(3+ln(x))^2/ln(x^5-6*x^4+17*x^3-25*x^2+20*x)^2
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {e^6 x^2 \left (-40+100 x-102 x^2+48 x^3-10 x^4+\left (40-50 x+34 x^2-12 x^3+2 x^4\right ) \log \left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )\right )}{\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right ) \log ^3\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )} \, dx=\frac {x^{2} e^{6}}{\log \left (x^{5} - 6 \, x^{4} + 17 \, x^{3} - 25 \, x^{2} + 20 \, x\right )^{2}} \] Input:

integrate(((2*x^4-12*x^3+34*x^2-50*x+40)*log(x^5-6*x^4+17*x^3-25*x^2+20*x) 
-10*x^4+48*x^3-102*x^2+100*x-40)*exp(3+log(x))^2/(x^5-6*x^4+17*x^3-25*x^2+ 
20*x)/log(x^5-6*x^4+17*x^3-25*x^2+20*x)^3,x, algorithm="fricas")
 

Output:

x^2*e^6/log(x^5 - 6*x^4 + 17*x^3 - 25*x^2 + 20*x)^2
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int \frac {e^6 x^2 \left (-40+100 x-102 x^2+48 x^3-10 x^4+\left (40-50 x+34 x^2-12 x^3+2 x^4\right ) \log \left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )\right )}{\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right ) \log ^3\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )} \, dx=\frac {x^{2} e^{6}}{\log {\left (x^{5} - 6 x^{4} + 17 x^{3} - 25 x^{2} + 20 x \right )}^{2}} \] Input:

integrate(((2*x**4-12*x**3+34*x**2-50*x+40)*ln(x**5-6*x**4+17*x**3-25*x**2 
+20*x)-10*x**4+48*x**3-102*x**2+100*x-40)*exp(3+ln(x))**2/(x**5-6*x**4+17* 
x**3-25*x**2+20*x)/ln(x**5-6*x**4+17*x**3-25*x**2+20*x)**3,x)
 

Output:

x**2*exp(6)/log(x**5 - 6*x**4 + 17*x**3 - 25*x**2 + 20*x)**2
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (27) = 54\).

Time = 0.08 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {e^6 x^2 \left (-40+100 x-102 x^2+48 x^3-10 x^4+\left (40-50 x+34 x^2-12 x^3+2 x^4\right ) \log \left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )\right )}{\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right ) \log ^3\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )} \, dx=\frac {x^{2} e^{6}}{\log \left (x^{4} - 6 \, x^{3} + 17 \, x^{2} - 25 \, x + 20\right )^{2} + 2 \, \log \left (x^{4} - 6 \, x^{3} + 17 \, x^{2} - 25 \, x + 20\right ) \log \left (x\right ) + \log \left (x\right )^{2}} \] Input:

integrate(((2*x^4-12*x^3+34*x^2-50*x+40)*log(x^5-6*x^4+17*x^3-25*x^2+20*x) 
-10*x^4+48*x^3-102*x^2+100*x-40)*exp(3+log(x))^2/(x^5-6*x^4+17*x^3-25*x^2+ 
20*x)/log(x^5-6*x^4+17*x^3-25*x^2+20*x)^3,x, algorithm="maxima")
 

Output:

x^2*e^6/(log(x^4 - 6*x^3 + 17*x^2 - 25*x + 20)^2 + 2*log(x^4 - 6*x^3 + 17* 
x^2 - 25*x + 20)*log(x) + log(x)^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (27) = 54\).

Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.19 \[ \int \frac {e^6 x^2 \left (-40+100 x-102 x^2+48 x^3-10 x^4+\left (40-50 x+34 x^2-12 x^3+2 x^4\right ) \log \left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )\right )}{\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right ) \log ^3\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )} \, dx=\frac {x^{2} e^{6}}{\log \left (x^{4} - 6 \, x^{3} + 17 \, x^{2} - 25 \, x + 20\right )^{2} + 2 \, \log \left (x^{4} - 6 \, x^{3} + 17 \, x^{2} - 25 \, x + 20\right ) \log \left (x\right ) + \log \left (x\right )^{2}} \] Input:

integrate(((2*x^4-12*x^3+34*x^2-50*x+40)*log(x^5-6*x^4+17*x^3-25*x^2+20*x) 
-10*x^4+48*x^3-102*x^2+100*x-40)*exp(3+log(x))^2/(x^5-6*x^4+17*x^3-25*x^2+ 
20*x)/log(x^5-6*x^4+17*x^3-25*x^2+20*x)^3,x, algorithm="giac")
 

Output:

x^2*e^6/(log(x^4 - 6*x^3 + 17*x^2 - 25*x + 20)^2 + 2*log(x^4 - 6*x^3 + 17* 
x^2 - 25*x + 20)*log(x) + log(x)^2)
 

Mupad [B] (verification not implemented)

Time = 4.01 (sec) , antiderivative size = 443, normalized size of antiderivative = 17.04 \[ \int \frac {e^6 x^2 \left (-40+100 x-102 x^2+48 x^3-10 x^4+\left (40-50 x+34 x^2-12 x^3+2 x^4\right ) \log \left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )\right )}{\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right ) \log ^3\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )} \, dx=\frac {x^2\,{\mathrm {e}}^6-\frac {x^2\,\ln \left (x^5-6\,x^4+17\,x^3-25\,x^2+20\,x\right )\,{\mathrm {e}}^6\,\left (x^4-6\,x^3+17\,x^2-25\,x+20\right )}{5\,x^4-24\,x^3+51\,x^2-50\,x+20}}{{\ln \left (x^5-6\,x^4+17\,x^3-25\,x^2+20\,x\right )}^2}-\frac {18\,x\,{\mathrm {e}}^6}{125}+\frac {2\,x^2\,{\mathrm {e}}^6}{25}+\frac {\frac {x^2\,{\mathrm {e}}^6\,\left (x^4-6\,x^3+17\,x^2-25\,x+20\right )}{5\,x^4-24\,x^3+51\,x^2-50\,x+20}-\frac {x^2\,\ln \left (x^5-6\,x^4+17\,x^3-25\,x^2+20\,x\right )\,{\mathrm {e}}^6\,\left (x^4-6\,x^3+17\,x^2-25\,x+20\right )\,\left (10\,x^8-102\,x^7+492\,x^6-1451\,x^5+2854\,x^4-3945\,x^3+3860\,x^2-2500\,x+800\right )}{{\left (5\,x^4-24\,x^3+51\,x^2-50\,x+20\right )}^3}}{\ln \left (x^5-6\,x^4+17\,x^3-25\,x^2+20\,x\right )}-\frac {-615\,{\mathrm {e}}^6\,x^{11}+8856\,{\mathrm {e}}^6\,x^{10}-\frac {276842\,{\mathrm {e}}^6\,x^9}{5}+\frac {1073739\,{\mathrm {e}}^6\,x^8}{5}-\frac {2975451\,{\mathrm {e}}^6\,x^7}{5}+\frac {6478543\,{\mathrm {e}}^6\,x^6}{5}-2302308\,{\mathrm {e}}^6\,x^5+3344556\,{\mathrm {e}}^6\,x^4-3744080\,{\mathrm {e}}^6\,x^3+2877120\,{\mathrm {e}}^6\,x^2-1200000\,{\mathrm {e}}^6\,x+140800\,{\mathrm {e}}^6}{15625\,x^{12}-225000\,x^{11}+1558125\,x^{10}-6786750\,x^9+20580375\,x^8-45571500\,x^7+75313875\,x^6-93378750\,x^5+86070000\,x^4-57475000\,x^3+26400000\,x^2-7500000\,x+1000000} \] Input:

int((exp(2*log(x) + 6)*(100*x + log(20*x - 25*x^2 + 17*x^3 - 6*x^4 + x^5)* 
(34*x^2 - 50*x - 12*x^3 + 2*x^4 + 40) - 102*x^2 + 48*x^3 - 10*x^4 - 40))/( 
log(20*x - 25*x^2 + 17*x^3 - 6*x^4 + x^5)^3*(20*x - 25*x^2 + 17*x^3 - 6*x^ 
4 + x^5)),x)
 

Output:

(x^2*exp(6) - (x^2*log(20*x - 25*x^2 + 17*x^3 - 6*x^4 + x^5)*exp(6)*(17*x^ 
2 - 25*x - 6*x^3 + x^4 + 20))/(51*x^2 - 50*x - 24*x^3 + 5*x^4 + 20))/log(2 
0*x - 25*x^2 + 17*x^3 - 6*x^4 + x^5)^2 - (18*x*exp(6))/125 + (2*x^2*exp(6) 
)/25 + ((x^2*exp(6)*(17*x^2 - 25*x - 6*x^3 + x^4 + 20))/(51*x^2 - 50*x - 2 
4*x^3 + 5*x^4 + 20) - (x^2*log(20*x - 25*x^2 + 17*x^3 - 6*x^4 + x^5)*exp(6 
)*(17*x^2 - 25*x - 6*x^3 + x^4 + 20)*(3860*x^2 - 2500*x - 3945*x^3 + 2854* 
x^4 - 1451*x^5 + 492*x^6 - 102*x^7 + 10*x^8 + 800))/(51*x^2 - 50*x - 24*x^ 
3 + 5*x^4 + 20)^3)/log(20*x - 25*x^2 + 17*x^3 - 6*x^4 + x^5) - (140800*exp 
(6) - 1200000*x*exp(6) + 2877120*x^2*exp(6) - 3744080*x^3*exp(6) + 3344556 
*x^4*exp(6) - 2302308*x^5*exp(6) + (6478543*x^6*exp(6))/5 - (2975451*x^7*e 
xp(6))/5 + (1073739*x^8*exp(6))/5 - (276842*x^9*exp(6))/5 + 8856*x^10*exp( 
6) - 615*x^11*exp(6))/(26400000*x^2 - 7500000*x - 57475000*x^3 + 86070000* 
x^4 - 93378750*x^5 + 75313875*x^6 - 45571500*x^7 + 20580375*x^8 - 6786750* 
x^9 + 1558125*x^10 - 225000*x^11 + 15625*x^12 + 1000000)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {e^6 x^2 \left (-40+100 x-102 x^2+48 x^3-10 x^4+\left (40-50 x+34 x^2-12 x^3+2 x^4\right ) \log \left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )\right )}{\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right ) \log ^3\left (20 x-25 x^2+17 x^3-6 x^4+x^5\right )} \, dx=\frac {e^{6} x^{2}}{\mathrm {log}\left (x^{5}-6 x^{4}+17 x^{3}-25 x^{2}+20 x \right )^{2}} \] Input:

int(((2*x^4-12*x^3+34*x^2-50*x+40)*log(x^5-6*x^4+17*x^3-25*x^2+20*x)-10*x^ 
4+48*x^3-102*x^2+100*x-40)*exp(3+log(x))^2/(x^5-6*x^4+17*x^3-25*x^2+20*x)/ 
log(x^5-6*x^4+17*x^3-25*x^2+20*x)^3,x)
 

Output:

(e**6*x**2)/log(x**5 - 6*x**4 + 17*x**3 - 25*x**2 + 20*x)**2