3.8.62 \(\int \frac {-2000-600 x-80 x^2+12 x^3+e^2 (-4500-1420 x-84 x^2+8 x^3)+(2000+700 x+40 x^2-4 x^3) \log (\frac {1}{4} (-20+x))}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+(1000+2550 x+750 x^2+36 x^3-4 x^4) \log (\frac {1}{4} (-20+x))+(-500-175 x-10 x^2+x^3) \log ^2(\frac {1}{4} (-20+x))} \, dx\) [762]

3.8.62.1 Optimal result

Integrand size = 139, antiderivative size = 33 \begin {dmath*} \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4 \left (-e^2+x\right )}{2+2 x-\frac {5}{5+x}-\log \left (-5+\frac {x}{4}\right )} \end {dmath*}

output

4*(x-exp(2))/(2*x-5/(5+x)-ln(1/4*x-5)+2)
 

3.8.62.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \begin {dmath*} \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4 (5+x) \left (-e^2+x\right )}{5+12 x+2 x^2-(5+x) \log \left (-5+\frac {x}{4}\right )} \end {dmath*}

input

Integrate[(-2000 - 600*x - 80*x^2 + 12*x^3 + E^2*(-4500 - 1420*x - 84*x^2 
+ 8*x^3) + (2000 + 700*x + 40*x^2 - 4*x^3)*Log[(-20 + x)/4])/(-500 - 2375* 
x - 3160*x^2 - 796*x^3 - 32*x^4 + 4*x^5 + (1000 + 2550*x + 750*x^2 + 36*x^ 
3 - 4*x^4)*Log[(-20 + x)/4] + (-500 - 175*x - 10*x^2 + x^3)*Log[(-20 + x)/ 
4]^2),x]
 

output

(4*(5 + x)*(-E^2 + x))/(5 + 12*x + 2*x^2 - (5 + x)*Log[-5 + x/4])
 

3.8.62.3 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.

\(\displaystyle \int \frac {12 x^3-80 x^2+e^2 \left (8 x^3-84 x^2-1420 x-4500\right )+\left (-4 x^3+40 x^2+700 x+2000\right ) \log \left (\frac {x-20}{4}\right )-600 x-2000}{4 x^5-32 x^4-796 x^3-3160 x^2+\left (x^3-10 x^2-175 x-500\right ) \log ^2\left (\frac {x-20}{4}\right )+\left (-4 x^4+36 x^3+750 x^2+2550 x+1000\right ) \log \left (\frac {x-20}{4}\right )-2375 x-500} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-12 x^3+80 x^2-e^2 \left (8 x^3-84 x^2-1420 x-4500\right )-\left (-4 x^3+40 x^2+700 x+2000\right ) \log \left (\frac {x-20}{4}\right )+600 x+2000}{(20-x) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (-\frac {80 x^2}{(x-20) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}-\frac {600 x}{(x-20) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}-\frac {4 (x+5)^2 \log \left (\frac {x}{4}-5\right )}{\left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}-\frac {2000}{(x-20) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}+\frac {12 x^3}{(x-20) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}+\frac {4 e^2 \left (2 x^3-21 x^2-355 x-1125\right )}{(x-20) \left (2 x^2+12 x-x \log \left (\frac {x}{4}-5\right )-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 100 e^2 \int \frac {1}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx+2500 \int \frac {1}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx-2500 e^2 \int \frac {1}{(x-20) \left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx+50000 \int \frac {1}{(x-20) \left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx+76 e^2 \int \frac {x}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx-100 \int \frac {x}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx+8 e^2 \int \frac {x^2}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx-76 \int \frac {x^2}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx+20 \int \frac {1}{2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5}dx+4 \int \frac {x}{2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5}dx-8 \int \frac {x^3}{\left (2 x^2-\log \left (\frac {x}{4}-5\right ) x+12 x-5 \log \left (\frac {x}{4}-5\right )+5\right )^2}dx\)

input

Int[(-2000 - 600*x - 80*x^2 + 12*x^3 + E^2*(-4500 - 1420*x - 84*x^2 + 8*x^ 
3) + (2000 + 700*x + 40*x^2 - 4*x^3)*Log[(-20 + x)/4])/(-500 - 2375*x - 31 
60*x^2 - 796*x^3 - 32*x^4 + 4*x^5 + (1000 + 2550*x + 750*x^2 + 36*x^3 - 4* 
x^4)*Log[(-20 + x)/4] + (-500 - 175*x - 10*x^2 + x^3)*Log[(-20 + x)/4]^2), 
x]
 

output

$Aborted
 

3.8.62.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.8.62.4 Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24

method	result	size
risch	\(-\frac {4 \left ({\mathrm e}^{2}-x \right ) \left (5+x \right )}{2 x^{2}-\ln \left (\frac {x}{4}-5\right ) x +12 x -5 \ln \left (\frac {x}{4}-5\right )+5}\)	\(41\)
parallelrisch	\(-\frac {4 \,{\mathrm e}^{2} x -4 x^{2}+20 \,{\mathrm e}^{2}-20 x}{2 x^{2}-\ln \left (\frac {x}{4}-5\right ) x +12 x -5 \ln \left (\frac {x}{4}-5\right )+5}\)	\(50\)
norman	\(\frac {\left (-4 \,{\mathrm e}^{2}-4\right ) x +10 \ln \left (\frac {x}{4}-5\right )-10+2 \ln \left (\frac {x}{4}-5\right ) x -20 \,{\mathrm e}^{2}}{2 x^{2}-\ln \left (\frac {x}{4}-5\right ) x +12 x -5 \ln \left (\frac {x}{4}-5\right )+5}\)	\(62\)
derivativedivides	\(\frac {-64 \left (\frac {x}{4}-5\right )^{2}+4 \left (-180+4 \,{\mathrm e}^{2}\right ) \left (\frac {x}{4}-5\right )-2000+100 \,{\mathrm e}^{2}}{4 \ln \left (\frac {x}{4}-5\right ) \left (\frac {x}{4}-5\right )-32 \left (\frac {x}{4}-5\right )^{2}+25 \ln \left (\frac {x}{4}-5\right )-92 x +795}\)	\(67\)
default	\(\frac {-64 \left (\frac {x}{4}-5\right )^{2}+4 \left (-180+4 \,{\mathrm e}^{2}\right ) \left (\frac {x}{4}-5\right )-2000+100 \,{\mathrm e}^{2}}{4 \ln \left (\frac {x}{4}-5\right ) \left (\frac {x}{4}-5\right )-32 \left (\frac {x}{4}-5\right )^{2}+25 \ln \left (\frac {x}{4}-5\right )-92 x +795}\)	\(67\)

input

int(((-4*x^3+40*x^2+700*x+2000)*ln(1/4*x-5)+(8*x^3-84*x^2-1420*x-4500)*exp 
(2)+12*x^3-80*x^2-600*x-2000)/((x^3-10*x^2-175*x-500)*ln(1/4*x-5)^2+(-4*x^ 
4+36*x^3+750*x^2+2550*x+1000)*ln(1/4*x-5)+4*x^5-32*x^4-796*x^3-3160*x^2-23 
75*x-500),x,method=_RETURNVERBOSE)
 

output

-4*(exp(2)-x)*(5+x)/(2*x^2-ln(1/4*x-5)*x+12*x-5*ln(1/4*x-5)+5)
 

3.8.62.5 Fricas [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \begin {dmath*} \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4 \, {\left (x^{2} - {\left (x + 5\right )} e^{2} + 5 \, x\right )}}{2 \, x^{2} - {\left (x + 5\right )} \log \left (\frac {1}{4} \, x - 5\right ) + 12 \, x + 5} \end {dmath*}

input

integrate(((-4*x^3+40*x^2+700*x+2000)*log(1/4*x-5)+(8*x^3-84*x^2-1420*x-45 
00)*exp(2)+12*x^3-80*x^2-600*x-2000)/((x^3-10*x^2-175*x-500)*log(1/4*x-5)^ 
2+(-4*x^4+36*x^3+750*x^2+2550*x+1000)*log(1/4*x-5)+4*x^5-32*x^4-796*x^3-31 
60*x^2-2375*x-500),x, algorithm=\
 

output

4*(x^2 - (x + 5)*e^2 + 5*x)/(2*x^2 - (x + 5)*log(1/4*x - 5) + 12*x + 5)
 

3.8.62.6 Sympy [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \begin {dmath*} \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {- 4 x^{2} - 20 x + 4 x e^{2} + 20 e^{2}}{- 2 x^{2} - 12 x + \left (x + 5\right ) \log {\left (\frac {x}{4} - 5 \right )} - 5} \end {dmath*}

input

integrate(((-4*x**3+40*x**2+700*x+2000)*ln(1/4*x-5)+(8*x**3-84*x**2-1420*x 
-4500)*exp(2)+12*x**3-80*x**2-600*x-2000)/((x**3-10*x**2-175*x-500)*ln(1/4 
*x-5)**2+(-4*x**4+36*x**3+750*x**2+2550*x+1000)*ln(1/4*x-5)+4*x**5-32*x**4 
-796*x**3-3160*x**2-2375*x-500),x)
 

output

(-4*x**2 - 20*x + 4*x*exp(2) + 20*exp(2))/(-2*x**2 - 12*x + (x + 5)*log(x/ 
4 - 5) - 5)
 

3.8.62.7 Maxima [A] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \begin {dmath*} \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4 \, {\left (x^{2} - x {\left (e^{2} - 5\right )} - 5 \, e^{2}\right )}}{2 \, x^{2} + 2 \, x {\left (\log \left (2\right ) + 6\right )} - {\left (x + 5\right )} \log \left (x - 20\right ) + 10 \, \log \left (2\right ) + 5} \end {dmath*}

input

integrate(((-4*x^3+40*x^2+700*x+2000)*log(1/4*x-5)+(8*x^3-84*x^2-1420*x-45 
00)*exp(2)+12*x^3-80*x^2-600*x-2000)/((x^3-10*x^2-175*x-500)*log(1/4*x-5)^ 
2+(-4*x^4+36*x^3+750*x^2+2550*x+1000)*log(1/4*x-5)+4*x^5-32*x^4-796*x^3-31 
60*x^2-2375*x-500),x, algorithm=\
 

output

4*(x^2 - x*(e^2 - 5) - 5*e^2)/(2*x^2 + 2*x*(log(2) + 6) - (x + 5)*log(x - 
20) + 10*log(2) + 5)
 

3.8.62.8 Giac [A] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.42 \begin {dmath*} \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4 \, {\left (x^{2} - x e^{2} + 5 \, x - 5 \, e^{2}\right )}}{2 \, x^{2} - x \log \left (\frac {1}{4} \, x - 5\right ) + 12 \, x - 5 \, \log \left (\frac {1}{4} \, x - 5\right ) + 5} \end {dmath*}

input

integrate(((-4*x^3+40*x^2+700*x+2000)*log(1/4*x-5)+(8*x^3-84*x^2-1420*x-45 
00)*exp(2)+12*x^3-80*x^2-600*x-2000)/((x^3-10*x^2-175*x-500)*log(1/4*x-5)^ 
2+(-4*x^4+36*x^3+750*x^2+2550*x+1000)*log(1/4*x-5)+4*x^5-32*x^4-796*x^3-31 
60*x^2-2375*x-500),x, algorithm=\
 

output

4*(x^2 - x*e^2 + 5*x - 5*e^2)/(2*x^2 - x*log(1/4*x - 5) + 12*x - 5*log(1/4 
*x - 5) + 5)
 

3.8.62.9 Mupad [B] (verification not implemented)

Time = 15.43 (sec) , antiderivative size = 113, normalized size of antiderivative = 3.42 \begin {dmath*} \int \frac {-2000-600 x-80 x^2+12 x^3+e^2 \left (-4500-1420 x-84 x^2+8 x^3\right )+\left (2000+700 x+40 x^2-4 x^3\right ) \log \left (\frac {1}{4} (-20+x)\right )}{-500-2375 x-3160 x^2-796 x^3-32 x^4+4 x^5+\left (1000+2550 x+750 x^2+36 x^3-4 x^4\right ) \log \left (\frac {1}{4} (-20+x)\right )+\left (-500-175 x-10 x^2+x^3\right ) \log ^2\left (\frac {1}{4} (-20+x)\right )} \, dx=\frac {4\,\left (112500\,x-112500\,{\mathrm {e}}^2-52375\,x\,{\mathrm {e}}^2-6300\,x^2\,{\mathrm {e}}^2+240\,x^3\,{\mathrm {e}}^2+51\,x^4\,{\mathrm {e}}^2-2\,x^5\,{\mathrm {e}}^2+52375\,x^2+6300\,x^3-240\,x^4-51\,x^5+2\,x^6\right )}{\left (12\,x+2\,x^2-\ln \left (\frac {x}{4}-5\right )\,\left (x+5\right )+5\right )\,\left (2\,x^4-61\,x^3+65\,x^2+5975\,x+22500\right )} \end {dmath*}

input

int((600*x + exp(2)*(1420*x + 84*x^2 - 8*x^3 + 4500) - log(x/4 - 5)*(700*x 
 + 40*x^2 - 4*x^3 + 2000) + 80*x^2 - 12*x^3 + 2000)/(2375*x + log(x/4 - 5) 
^2*(175*x + 10*x^2 - x^3 + 500) + 3160*x^2 + 796*x^3 + 32*x^4 - 4*x^5 - lo 
g(x/4 - 5)*(2550*x + 750*x^2 + 36*x^3 - 4*x^4 + 1000) + 500),x)
 

output

(4*(112500*x - 112500*exp(2) - 52375*x*exp(2) - 6300*x^2*exp(2) + 240*x^3* 
exp(2) + 51*x^4*exp(2) - 2*x^5*exp(2) + 52375*x^2 + 6300*x^3 - 240*x^4 - 5 
1*x^5 + 2*x^6))/((12*x + 2*x^2 - log(x/4 - 5)*(x + 5) + 5)*(5975*x + 65*x^ 
2 - 61*x^3 + 2*x^4 + 22500))