\(\int \frac {-e^9-2100 x-275 x^2+(-12250 x-1500 x^2) \log (x)+(-8750 x-625 x^2) \log ^2(x)}{e^{18}-50 e^9 x^2+625 x^4+(-500 e^9 x^2+12500 x^4) \log (x)+(-1250 e^9 x^2+93750 x^4) \log ^2(x)+312500 x^4 \log ^3(x)+390625 x^4 \log ^4(x)} \, dx\) [1701]

Optimal result

Integrand size = 114, antiderivative size = 23 \[ \int \frac {-e^9-2100 x-275 x^2+\left (-12250 x-1500 x^2\right ) \log (x)+\left (-8750 x-625 x^2\right ) \log ^2(x)}{e^{18}-50 e^9 x^2+625 x^4+\left (-500 e^9 x^2+12500 x^4\right ) \log (x)+\left (-1250 e^9 x^2+93750 x^4\right ) \log ^2(x)+312500 x^4 \log ^3(x)+390625 x^4 \log ^4(x)} \, dx=\frac {7+x}{-e^9+25 (x+5 x \log (x))^2} \] Output:

(7+x)/(5*(x+5*x*ln(x))*(5*x+25*x*ln(x))-exp(9))
 

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-e^9-2100 x-275 x^2+\left (-12250 x-1500 x^2\right ) \log (x)+\left (-8750 x-625 x^2\right ) \log ^2(x)}{e^{18}-50 e^9 x^2+625 x^4+\left (-500 e^9 x^2+12500 x^4\right ) \log (x)+\left (-1250 e^9 x^2+93750 x^4\right ) \log ^2(x)+312500 x^4 \log ^3(x)+390625 x^4 \log ^4(x)} \, dx=-\frac {7+x}{e^9-25 x^2-250 x^2 \log (x)-625 x^2 \log ^2(x)} \] Input:

Integrate[(-E^9 - 2100*x - 275*x^2 + (-12250*x - 1500*x^2)*Log[x] + (-8750 
*x - 625*x^2)*Log[x]^2)/(E^18 - 50*E^9*x^2 + 625*x^4 + (-500*E^9*x^2 + 125 
00*x^4)*Log[x] + (-1250*E^9*x^2 + 93750*x^4)*Log[x]^2 + 312500*x^4*Log[x]^ 
3 + 390625*x^4*Log[x]^4),x]
 

Output:

-((7 + x)/(E^9 - 25*x^2 - 250*x^2*Log[x] - 625*x^2*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.

\(\displaystyle \int \frac {-275 x^2+\left (-625 x^2-8750 x\right ) \log ^2(x)+\left (-1500 x^2-12250 x\right ) \log (x)-2100 x-e^9}{625 x^4+390625 x^4 \log ^4(x)+312500 x^4 \log ^3(x)-50 e^9 x^2+\left (93750 x^4-1250 e^9 x^2\right ) \log ^2(x)+\left (12500 x^4-500 e^9 x^2\right ) \log (x)+e^{18}} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-25 x (11 x+84)-625 x (x+14) \log ^2(x)-250 x (6 x+49) \log (x)-e^9}{\left (-25 x^2-625 x^2 \log ^2(x)-250 x^2 \log (x)+e^9\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {x+14}{x \left (-25 x^2-625 x^2 \log ^2(x)-250 x^2 \log (x)+e^9\right )}-\frac {2 (x+7) \left (125 x^2+625 x^2 \log (x)+e^9\right )}{x \left (25 x^2+625 x^2 \log ^2(x)+250 x^2 \log (x)-e^9\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -2 e^9 \int \frac {1}{\left (-625 \log ^2(x) x^2-250 \log (x) x^2-25 x^2+e^9\right )^2}dx-1750 \int \frac {x}{\left (-625 \log ^2(x) x^2-250 \log (x) x^2-25 x^2+e^9\right )^2}dx-250 \int \frac {x^2}{\left (-625 \log ^2(x) x^2-250 \log (x) x^2-25 x^2+e^9\right )^2}dx-8750 \int \frac {x \log (x)}{\left (-625 \log ^2(x) x^2-250 \log (x) x^2-25 x^2+e^9\right )^2}dx-1250 \int \frac {x^2 \log (x)}{\left (-625 \log ^2(x) x^2-250 \log (x) x^2-25 x^2+e^9\right )^2}dx+\int \frac {1}{-625 \log ^2(x) x^2-250 \log (x) x^2-25 x^2+e^9}dx+14 \int \frac {1}{x \left (-625 \log ^2(x) x^2-250 \log (x) x^2-25 x^2+e^9\right )}dx-14 e^9 \int \frac {1}{x \left (625 \log ^2(x) x^2+250 \log (x) x^2+25 x^2-e^9\right )^2}dx\)

Input:

Int[(-E^9 - 2100*x - 275*x^2 + (-12250*x - 1500*x^2)*Log[x] + (-8750*x - 6 
25*x^2)*Log[x]^2)/(E^18 - 50*E^9*x^2 + 625*x^4 + (-500*E^9*x^2 + 12500*x^4 
)*Log[x] + (-1250*E^9*x^2 + 93750*x^4)*Log[x]^2 + 312500*x^4*Log[x]^3 + 39 
0625*x^4*Log[x]^4),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39

method	result	size
default	\(-\frac {x +7}{-625 x^{2} \ln \left (x \right )^{2}-250 x^{2} \ln \left (x \right )-25 x^{2}+{\mathrm e}^{9}}\)	\(32\)
risch	\(-\frac {x +7}{-625 x^{2} \ln \left (x \right )^{2}-250 x^{2} \ln \left (x \right )-25 x^{2}+{\mathrm e}^{9}}\)	\(32\)
norman	\(\frac {-x -7}{-625 x^{2} \ln \left (x \right )^{2}-250 x^{2} \ln \left (x \right )-25 x^{2}+{\mathrm e}^{9}}\)	\(33\)
parallelrisch	\(\frac {-175-25 x}{-15625 x^{2} \ln \left (x \right )^{2}-6250 x^{2} \ln \left (x \right )-625 x^{2}+25 \,{\mathrm e}^{9}}\)	\(34\)

Input:

int(((-625*x^2-8750*x)*ln(x)^2+(-1500*x^2-12250*x)*ln(x)-exp(9)-275*x^2-21 
00*x)/(390625*x^4*ln(x)^4+312500*x^4*ln(x)^3+(-1250*x^2*exp(9)+93750*x^4)* 
ln(x)^2+(-500*x^2*exp(9)+12500*x^4)*ln(x)+exp(9)^2-50*x^2*exp(9)+625*x^4), 
x,method=_RETURNVERBOSE)
 

Output:

-(x+7)/(-625*x^2*ln(x)^2-250*x^2*ln(x)-25*x^2+exp(9))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-e^9-2100 x-275 x^2+\left (-12250 x-1500 x^2\right ) \log (x)+\left (-8750 x-625 x^2\right ) \log ^2(x)}{e^{18}-50 e^9 x^2+625 x^4+\left (-500 e^9 x^2+12500 x^4\right ) \log (x)+\left (-1250 e^9 x^2+93750 x^4\right ) \log ^2(x)+312500 x^4 \log ^3(x)+390625 x^4 \log ^4(x)} \, dx=\frac {x + 7}{625 \, x^{2} \log \left (x\right )^{2} + 250 \, x^{2} \log \left (x\right ) + 25 \, x^{2} - e^{9}} \] Input:

integrate(((-625*x^2-8750*x)*log(x)^2+(-1500*x^2-12250*x)*log(x)-exp(9)-27 
5*x^2-2100*x)/(390625*x^4*log(x)^4+312500*x^4*log(x)^3+(-1250*x^2*exp(9)+9 
3750*x^4)*log(x)^2+(-500*x^2*exp(9)+12500*x^4)*log(x)+exp(9)^2-50*x^2*exp( 
9)+625*x^4),x, algorithm="fricas")
 

Output:

(x + 7)/(625*x^2*log(x)^2 + 250*x^2*log(x) + 25*x^2 - e^9)
 

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {-e^9-2100 x-275 x^2+\left (-12250 x-1500 x^2\right ) \log (x)+\left (-8750 x-625 x^2\right ) \log ^2(x)}{e^{18}-50 e^9 x^2+625 x^4+\left (-500 e^9 x^2+12500 x^4\right ) \log (x)+\left (-1250 e^9 x^2+93750 x^4\right ) \log ^2(x)+312500 x^4 \log ^3(x)+390625 x^4 \log ^4(x)} \, dx=\frac {x + 7}{625 x^{2} \log {\left (x \right )}^{2} + 250 x^{2} \log {\left (x \right )} + 25 x^{2} - e^{9}} \] Input:

integrate(((-625*x**2-8750*x)*ln(x)**2+(-1500*x**2-12250*x)*ln(x)-exp(9)-2 
75*x**2-2100*x)/(390625*x**4*ln(x)**4+312500*x**4*ln(x)**3+(-1250*x**2*exp 
(9)+93750*x**4)*ln(x)**2+(-500*x**2*exp(9)+12500*x**4)*ln(x)+exp(9)**2-50* 
x**2*exp(9)+625*x**4),x)
 

Output:

(x + 7)/(625*x**2*log(x)**2 + 250*x**2*log(x) + 25*x**2 - exp(9))
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.39 \[ \int \frac {-e^9-2100 x-275 x^2+\left (-12250 x-1500 x^2\right ) \log (x)+\left (-8750 x-625 x^2\right ) \log ^2(x)}{e^{18}-50 e^9 x^2+625 x^4+\left (-500 e^9 x^2+12500 x^4\right ) \log (x)+\left (-1250 e^9 x^2+93750 x^4\right ) \log ^2(x)+312500 x^4 \log ^3(x)+390625 x^4 \log ^4(x)} \, dx=\frac {x + 7}{625 \, x^{2} \log \left (x\right )^{2} + 250 \, x^{2} \log \left (x\right ) + 25 \, x^{2} - e^{9}} \] Input:

integrate(((-625*x^2-8750*x)*log(x)^2+(-1500*x^2-12250*x)*log(x)-exp(9)-27 
5*x^2-2100*x)/(390625*x^4*log(x)^4+312500*x^4*log(x)^3+(-1250*x^2*exp(9)+9 
3750*x^4)*log(x)^2+(-500*x^2*exp(9)+12500*x^4)*log(x)+exp(9)^2-50*x^2*exp( 
9)+625*x^4),x, algorithm="maxima")
 

Output:

(x + 7)/(625*x^2*log(x)^2 + 250*x^2*log(x) + 25*x^2 - e^9)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {-e^9-2100 x-275 x^2+\left (-12250 x-1500 x^2\right ) \log (x)+\left (-8750 x-625 x^2\right ) \log ^2(x)}{e^{18}-50 e^9 x^2+625 x^4+\left (-500 e^9 x^2+12500 x^4\right ) \log (x)+\left (-1250 e^9 x^2+93750 x^4\right ) \log ^2(x)+312500 x^4 \log ^3(x)+390625 x^4 \log ^4(x)} \, dx=\frac {2 \, {\left (x + 7\right )}}{625 \, x^{2} \log \left (x\right )^{2} + 250 \, x^{2} \log \left (x\right ) + 25 \, x^{2} - e^{9}} \] Input:

integrate(((-625*x^2-8750*x)*log(x)^2+(-1500*x^2-12250*x)*log(x)-exp(9)-27 
5*x^2-2100*x)/(390625*x^4*log(x)^4+312500*x^4*log(x)^3+(-1250*x^2*exp(9)+9 
3750*x^4)*log(x)^2+(-500*x^2*exp(9)+12500*x^4)*log(x)+exp(9)^2-50*x^2*exp( 
9)+625*x^4),x, algorithm="giac")
 

Output:

2*(x + 7)/(625*x^2*log(x)^2 + 250*x^2*log(x) + 25*x^2 - e^9)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {-e^9-2100 x-275 x^2+\left (-12250 x-1500 x^2\right ) \log (x)+\left (-8750 x-625 x^2\right ) \log ^2(x)}{e^{18}-50 e^9 x^2+625 x^4+\left (-500 e^9 x^2+12500 x^4\right ) \log (x)+\left (-1250 e^9 x^2+93750 x^4\right ) \log ^2(x)+312500 x^4 \log ^3(x)+390625 x^4 \log ^4(x)} \, dx=\int -\frac {2100\,x+{\mathrm {e}}^9+{\ln \left (x\right )}^2\,\left (625\,x^2+8750\,x\right )+\ln \left (x\right )\,\left (1500\,x^2+12250\,x\right )+275\,x^2}{{\mathrm {e}}^{18}-\ln \left (x\right )\,\left (500\,x^2\,{\mathrm {e}}^9-12500\,x^4\right )+312500\,x^4\,{\ln \left (x\right )}^3+390625\,x^4\,{\ln \left (x\right )}^4-50\,x^2\,{\mathrm {e}}^9-{\ln \left (x\right )}^2\,\left (1250\,x^2\,{\mathrm {e}}^9-93750\,x^4\right )+625\,x^4} \,d x \] Input:

int(-(2100*x + exp(9) + log(x)^2*(8750*x + 625*x^2) + log(x)*(12250*x + 15 
00*x^2) + 275*x^2)/(exp(18) - log(x)*(500*x^2*exp(9) - 12500*x^4) + 312500 
*x^4*log(x)^3 + 390625*x^4*log(x)^4 - 50*x^2*exp(9) - log(x)^2*(1250*x^2*e 
xp(9) - 93750*x^4) + 625*x^4),x)
 

Output:

int(-(2100*x + exp(9) + log(x)^2*(8750*x + 625*x^2) + log(x)*(12250*x + 15 
00*x^2) + 275*x^2)/(exp(18) - log(x)*(500*x^2*exp(9) - 12500*x^4) + 312500 
*x^4*log(x)^3 + 390625*x^4*log(x)^4 - 50*x^2*exp(9) - log(x)^2*(1250*x^2*e 
xp(9) - 93750*x^4) + 625*x^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {-e^9-2100 x-275 x^2+\left (-12250 x-1500 x^2\right ) \log (x)+\left (-8750 x-625 x^2\right ) \log ^2(x)}{e^{18}-50 e^9 x^2+625 x^4+\left (-500 e^9 x^2+12500 x^4\right ) \log (x)+\left (-1250 e^9 x^2+93750 x^4\right ) \log ^2(x)+312500 x^4 \log ^3(x)+390625 x^4 \log ^4(x)} \, dx=\frac {x +7}{625 \mathrm {log}\left (x \right )^{2} x^{2}+250 \,\mathrm {log}\left (x \right ) x^{2}-e^{9}+25 x^{2}} \] Input:

int(((-625*x^2-8750*x)*log(x)^2+(-1500*x^2-12250*x)*log(x)-exp(9)-275*x^2- 
2100*x)/(390625*x^4*log(x)^4+312500*x^4*log(x)^3+(-1250*x^2*exp(9)+93750*x 
^4)*log(x)^2+(-500*x^2*exp(9)+12500*x^4)*log(x)+exp(9)^2-50*x^2*exp(9)+625 
*x^4),x)
 

Output:

(x + 7)/(625*log(x)**2*x**2 + 250*log(x)*x**2 - e**9 + 25*x**2)