\(\int \frac {(-x^3-x^4) \log (x)+(5 x^2+6 x^3+x^4+(5 x^2+6 x^3+x^4) \log (x)) \log (5+x)+(-500-600 x-100 x^2) \log ^2(5+x)+((5 x^2+11 x^3+2 x^4) \log (x) \log (5+x)+(500+1225 x+475 x^2+50 x^3) \log ^2(5+x)) \log (\frac {x^2 \log (x)+(100+25 x) \log (5+x)}{25 x \log (5+x)})}{(5 x^2+x^3) \log (x) \log (5+x)+(500+225 x+25 x^2) \log ^2(5+x)} \, dx\) [1480]

Optimal result

Integrand size = 183, antiderivative size = 28 \[ \int \frac {\left (-x^3-x^4\right ) \log (x)+\left (5 x^2+6 x^3+x^4+\left (5 x^2+6 x^3+x^4\right ) \log (x)\right ) \log (5+x)+\left (-500-600 x-100 x^2\right ) \log ^2(5+x)+\left (\left (5 x^2+11 x^3+2 x^4\right ) \log (x) \log (5+x)+\left (500+1225 x+475 x^2+50 x^3\right ) \log ^2(5+x)\right ) \log \left (\frac {x^2 \log (x)+(100+25 x) \log (5+x)}{25 x \log (5+x)}\right )}{\left (5 x^2+x^3\right ) \log (x) \log (5+x)+\left (500+225 x+25 x^2\right ) \log ^2(5+x)} \, dx=x (1+x) \log \left (\frac {4+x+\frac {x^2 \log (x)}{25 \log (5+x)}}{x}\right ) \] Output:

ln((4+1/25*x^2/ln(5+x)*ln(x)+x)/x)*(1+x)*x
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\left (-x^3-x^4\right ) \log (x)+\left (5 x^2+6 x^3+x^4+\left (5 x^2+6 x^3+x^4\right ) \log (x)\right ) \log (5+x)+\left (-500-600 x-100 x^2\right ) \log ^2(5+x)+\left (\left (5 x^2+11 x^3+2 x^4\right ) \log (x) \log (5+x)+\left (500+1225 x+475 x^2+50 x^3\right ) \log ^2(5+x)\right ) \log \left (\frac {x^2 \log (x)+(100+25 x) \log (5+x)}{25 x \log (5+x)}\right )}{\left (5 x^2+x^3\right ) \log (x) \log (5+x)+\left (500+225 x+25 x^2\right ) \log ^2(5+x)} \, dx=x (1+x) \log \left (1+\frac {4}{x}+\frac {x \log (x)}{25 \log (5+x)}\right ) \] Input:

Integrate[((-x^3 - x^4)*Log[x] + (5*x^2 + 6*x^3 + x^4 + (5*x^2 + 6*x^3 + x 
^4)*Log[x])*Log[5 + x] + (-500 - 600*x - 100*x^2)*Log[5 + x]^2 + ((5*x^2 + 
 11*x^3 + 2*x^4)*Log[x]*Log[5 + x] + (500 + 1225*x + 475*x^2 + 50*x^3)*Log 
[5 + x]^2)*Log[(x^2*Log[x] + (100 + 25*x)*Log[5 + x])/(25*x*Log[5 + x])])/ 
((5*x^2 + x^3)*Log[x]*Log[5 + x] + (500 + 225*x + 25*x^2)*Log[5 + x]^2),x]
 

Output:

x*(1 + x)*Log[1 + 4/x + (x*Log[x])/(25*Log[5 + x])]
 

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.

\(\displaystyle \int \frac {\left (-100 x^2-600 x-500\right ) \log ^2(x+5)+\left (-x^4-x^3\right ) \log (x)+\left (\left (50 x^3+475 x^2+1225 x+500\right ) \log ^2(x+5)+\left (2 x^4+11 x^3+5 x^2\right ) \log (x) \log (x+5)\right ) \log \left (\frac {x^2 \log (x)+(25 x+100) \log (x+5)}{25 x \log (x+5)}\right )+\left (x^4+6 x^3+5 x^2+\left (x^4+6 x^3+5 x^2\right ) \log (x)\right ) \log (x+5)}{\left (25 x^2+225 x+500\right ) \log ^2(x+5)+\left (x^3+5 x^2\right ) \log (x) \log (x+5)} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {\left (-100 x^2-600 x-500\right ) \log ^2(x+5)+\left (-x^4-x^3\right ) \log (x)+\left (\left (50 x^3+475 x^2+1225 x+500\right ) \log ^2(x+5)+\left (2 x^4+11 x^3+5 x^2\right ) \log (x) \log (x+5)\right ) \log \left (\frac {x^2 \log (x)+(25 x+100) \log (x+5)}{25 x \log (x+5)}\right )+\left (x^4+6 x^3+5 x^2+\left (x^4+6 x^3+5 x^2\right ) \log (x)\right ) \log (x+5)}{(x+5) \log (x+5) \left (x^2 \log (x)+25 x \log (x+5)+100 \log (x+5)\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {(x+1) \left (x^3 (-\log (x))+x^3 \log (x) \log (x+5)+x^3 \log (x+5)+5 x^2 \log (x) \log (x+5)+5 x^2 \log (x+5)-100 x \log ^2(x+5)-500 \log ^2(x+5)\right )}{(x+5) \log (x+5) \left (x^2 \log (x)+25 x \log (x+5)+100 \log (x+5)\right )}+(2 x+1) \log \left (\frac {4}{x}+\frac {x \log (x)}{25 \log (x+5)}+1\right )\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 100 \int \frac {1}{\log (x) x^2+25 \log (x+5) x+100 \log (x+5)}dx+26 \int \frac {x^2}{\log (x) x^2+25 \log (x+5) x+100 \log (x+5)}dx-500 \int \frac {1}{(x+5) \left (\log (x) x^2+25 \log (x+5) x+100 \log (x+5)\right )}dx+48 \int \frac {\log (x)}{\log (x) x^2+25 \log (x+5) x+100 \log (x+5)}dx-12 \int \frac {x \log (x)}{\log (x) x^2+25 \log (x+5) x+100 \log (x+5)}dx+5 \int \frac {x^2 \log (x)}{\log (x) x^2+25 \log (x+5) x+100 \log (x+5)}dx-192 \int \frac {\log (x)}{(x+4) \left (\log (x) x^2+25 \log (x+5) x+100 \log (x+5)\right )}dx+\int \frac {x^3}{\log (x) x^2+25 \log (x+5) x+100 \log (x+5)}dx+\int \frac {x^3 \log (x)}{\log (x) x^2+25 \log (x+5) x+100 \log (x+5)}dx+\int \log \left (\frac {x \log (x)}{25 \log (x+5)}+\frac {4}{x}+1\right )dx+2 \int x \log \left (\frac {x \log (x)}{25 \log (x+5)}+\frac {4}{x}+1\right )dx-\operatorname {ExpIntegralEi}(2 \log (x+5))+9 \operatorname {LogIntegral}(x+5)-4 x+12 \log (x+4)-20 \log (\log (x+5))\)

Input:

Int[((-x^3 - x^4)*Log[x] + (5*x^2 + 6*x^3 + x^4 + (5*x^2 + 6*x^3 + x^4)*Lo 
g[x])*Log[5 + x] + (-500 - 600*x - 100*x^2)*Log[5 + x]^2 + ((5*x^2 + 11*x^ 
3 + 2*x^4)*Log[x]*Log[5 + x] + (500 + 1225*x + 475*x^2 + 50*x^3)*Log[5 + x 
]^2)*Log[(x^2*Log[x] + (100 + 25*x)*Log[5 + x])/(25*x*Log[5 + x])])/((5*x^ 
2 + x^3)*Log[x]*Log[5 + x] + (500 + 225*x + 25*x^2)*Log[5 + x]^2),x]
 

Output:

$Aborted
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(65\) vs. \(2(26)=52\).

Time = 129.41 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.36

method	result	size
parallelrisch	\(\ln \left (\frac {\left (25 x +100\right ) \ln \left (5+x \right )+x^{2} \ln \left (x \right )}{25 x \ln \left (5+x \right )}\right ) x^{2}+\ln \left (\frac {\left (25 x +100\right ) \ln \left (5+x \right )+x^{2} \ln \left (x \right )}{25 x \ln \left (5+x \right )}\right ) x\)	\(66\)
risch	\(\text {Expression too large to display}\)	\(979\)

Input:

int((((50*x^3+475*x^2+1225*x+500)*ln(5+x)^2+(2*x^4+11*x^3+5*x^2)*ln(x)*ln( 
5+x))*ln(1/25*((25*x+100)*ln(5+x)+x^2*ln(x))/x/ln(5+x))+(-100*x^2-600*x-50 
0)*ln(5+x)^2+((x^4+6*x^3+5*x^2)*ln(x)+x^4+6*x^3+5*x^2)*ln(5+x)+(-x^4-x^3)* 
ln(x))/((25*x^2+225*x+500)*ln(5+x)^2+(x^3+5*x^2)*ln(x)*ln(5+x)),x,method=_ 
RETURNVERBOSE)
 

Output:

ln(1/25*((25*x+100)*ln(5+x)+x^2*ln(x))/x/ln(5+x))*x^2+ln(1/25*((25*x+100)* 
ln(5+x)+x^2*ln(x))/x/ln(5+x))*x
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.21 \[ \int \frac {\left (-x^3-x^4\right ) \log (x)+\left (5 x^2+6 x^3+x^4+\left (5 x^2+6 x^3+x^4\right ) \log (x)\right ) \log (5+x)+\left (-500-600 x-100 x^2\right ) \log ^2(5+x)+\left (\left (5 x^2+11 x^3+2 x^4\right ) \log (x) \log (5+x)+\left (500+1225 x+475 x^2+50 x^3\right ) \log ^2(5+x)\right ) \log \left (\frac {x^2 \log (x)+(100+25 x) \log (5+x)}{25 x \log (5+x)}\right )}{\left (5 x^2+x^3\right ) \log (x) \log (5+x)+\left (500+225 x+25 x^2\right ) \log ^2(5+x)} \, dx={\left (x^{2} + x\right )} \log \left (\frac {x^{2} \log \left (x\right ) + 25 \, {\left (x + 4\right )} \log \left (x + 5\right )}{25 \, x \log \left (x + 5\right )}\right ) \] Input:

integrate((((50*x^3+475*x^2+1225*x+500)*log(5+x)^2+(2*x^4+11*x^3+5*x^2)*lo 
g(x)*log(5+x))*log(1/25*((25*x+100)*log(5+x)+x^2*log(x))/x/log(5+x))+(-100 
*x^2-600*x-500)*log(5+x)^2+((x^4+6*x^3+5*x^2)*log(x)+x^4+6*x^3+5*x^2)*log( 
5+x)+(-x^4-x^3)*log(x))/((25*x^2+225*x+500)*log(5+x)^2+(x^3+5*x^2)*log(x)* 
log(5+x)),x, algorithm="fricas")
 

Output:

(x^2 + x)*log(1/25*(x^2*log(x) + 25*(x + 4)*log(x + 5))/(x*log(x + 5)))
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {\left (-x^3-x^4\right ) \log (x)+\left (5 x^2+6 x^3+x^4+\left (5 x^2+6 x^3+x^4\right ) \log (x)\right ) \log (5+x)+\left (-500-600 x-100 x^2\right ) \log ^2(5+x)+\left (\left (5 x^2+11 x^3+2 x^4\right ) \log (x) \log (5+x)+\left (500+1225 x+475 x^2+50 x^3\right ) \log ^2(5+x)\right ) \log \left (\frac {x^2 \log (x)+(100+25 x) \log (5+x)}{25 x \log (5+x)}\right )}{\left (5 x^2+x^3\right ) \log (x) \log (5+x)+\left (500+225 x+25 x^2\right ) \log ^2(5+x)} \, dx=\text {Exception raised: PolynomialError} \] Input:

integrate((((50*x**3+475*x**2+1225*x+500)*ln(5+x)**2+(2*x**4+11*x**3+5*x** 
2)*ln(x)*ln(5+x))*ln(1/25*((25*x+100)*ln(5+x)+x**2*ln(x))/x/ln(5+x))+(-100 
*x**2-600*x-500)*ln(5+x)**2+((x**4+6*x**3+5*x**2)*ln(x)+x**4+6*x**3+5*x**2 
)*ln(5+x)+(-x**4-x**3)*ln(x))/((25*x**2+225*x+500)*ln(5+x)**2+(x**3+5*x**2 
)*ln(x)*ln(5+x)),x)
 

Output:

Exception raised: PolynomialError >> 1/(75*x**3 + 975*x**2 + 4200*x + 6000 
) contains an element of the set of generators.
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (28) = 56\).

Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.18 \[ \int \frac {\left (-x^3-x^4\right ) \log (x)+\left (5 x^2+6 x^3+x^4+\left (5 x^2+6 x^3+x^4\right ) \log (x)\right ) \log (5+x)+\left (-500-600 x-100 x^2\right ) \log ^2(5+x)+\left (\left (5 x^2+11 x^3+2 x^4\right ) \log (x) \log (5+x)+\left (500+1225 x+475 x^2+50 x^3\right ) \log ^2(5+x)\right ) \log \left (\frac {x^2 \log (x)+(100+25 x) \log (5+x)}{25 x \log (5+x)}\right )}{\left (5 x^2+x^3\right ) \log (x) \log (5+x)+\left (500+225 x+25 x^2\right ) \log ^2(5+x)} \, dx=-2 \, x^{2} \log \left (5\right ) - 2 \, x \log \left (5\right ) + {\left (x^{2} + x\right )} \log \left (x^{2} \log \left (x\right ) + 25 \, x \log \left (x + 5\right ) + 100 \, \log \left (x + 5\right )\right ) - {\left (x^{2} + x\right )} \log \left (x\right ) - {\left (x^{2} + x\right )} \log \left (\log \left (x + 5\right )\right ) \] Input:

integrate((((50*x^3+475*x^2+1225*x+500)*log(5+x)^2+(2*x^4+11*x^3+5*x^2)*lo 
g(x)*log(5+x))*log(1/25*((25*x+100)*log(5+x)+x^2*log(x))/x/log(5+x))+(-100 
*x^2-600*x-500)*log(5+x)^2+((x^4+6*x^3+5*x^2)*log(x)+x^4+6*x^3+5*x^2)*log( 
5+x)+(-x^4-x^3)*log(x))/((25*x^2+225*x+500)*log(5+x)^2+(x^3+5*x^2)*log(x)* 
log(5+x)),x, algorithm="maxima")
 

Output:

-2*x^2*log(5) - 2*x*log(5) + (x^2 + x)*log(x^2*log(x) + 25*x*log(x + 5) + 
100*log(x + 5)) - (x^2 + x)*log(x) - (x^2 + x)*log(log(x + 5))
 

Giac [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.82 \[ \int \frac {\left (-x^3-x^4\right ) \log (x)+\left (5 x^2+6 x^3+x^4+\left (5 x^2+6 x^3+x^4\right ) \log (x)\right ) \log (5+x)+\left (-500-600 x-100 x^2\right ) \log ^2(5+x)+\left (\left (5 x^2+11 x^3+2 x^4\right ) \log (x) \log (5+x)+\left (500+1225 x+475 x^2+50 x^3\right ) \log ^2(5+x)\right ) \log \left (\frac {x^2 \log (x)+(100+25 x) \log (5+x)}{25 x \log (5+x)}\right )}{\left (5 x^2+x^3\right ) \log (x) \log (5+x)+\left (500+225 x+25 x^2\right ) \log ^2(5+x)} \, dx={\left (x^{2} + x\right )} \log \left (x^{2} \log \left (x\right ) + 25 \, x \log \left (x + 5\right ) + 100 \, \log \left (x + 5\right )\right ) - {\left (x^{2} + x\right )} \log \left (x\right ) - {\left (x^{2} + x\right )} \log \left (25 \, \log \left (x + 5\right )\right ) \] Input:

integrate((((50*x^3+475*x^2+1225*x+500)*log(5+x)^2+(2*x^4+11*x^3+5*x^2)*lo 
g(x)*log(5+x))*log(1/25*((25*x+100)*log(5+x)+x^2*log(x))/x/log(5+x))+(-100 
*x^2-600*x-500)*log(5+x)^2+((x^4+6*x^3+5*x^2)*log(x)+x^4+6*x^3+5*x^2)*log( 
5+x)+(-x^4-x^3)*log(x))/((25*x^2+225*x+500)*log(5+x)^2+(x^3+5*x^2)*log(x)* 
log(5+x)),x, algorithm="giac")
 

Output:

(x^2 + x)*log(x^2*log(x) + 25*x*log(x + 5) + 100*log(x + 5)) - (x^2 + x)*l 
og(x) - (x^2 + x)*log(25*log(x + 5))
 

Mupad [B] (verification not implemented)

Time = 4.13 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.25 \[ \int \frac {\left (-x^3-x^4\right ) \log (x)+\left (5 x^2+6 x^3+x^4+\left (5 x^2+6 x^3+x^4\right ) \log (x)\right ) \log (5+x)+\left (-500-600 x-100 x^2\right ) \log ^2(5+x)+\left (\left (5 x^2+11 x^3+2 x^4\right ) \log (x) \log (5+x)+\left (500+1225 x+475 x^2+50 x^3\right ) \log ^2(5+x)\right ) \log \left (\frac {x^2 \log (x)+(100+25 x) \log (5+x)}{25 x \log (5+x)}\right )}{\left (5 x^2+x^3\right ) \log (x) \log (5+x)+\left (500+225 x+25 x^2\right ) \log ^2(5+x)} \, dx=\frac {\ln \left (\frac {\frac {x^2\,\ln \left (x\right )}{25}+\frac {\ln \left (x+5\right )\,\left (25\,x+100\right )}{25}}{x\,\ln \left (x+5\right )}\right )\,\left (x^5+10\,x^4+29\,x^3+20\,x^2\right )}{x\,\left (x+4\right )\,\left (x+5\right )} \] Input:

int((log(x + 5)*(5*x^2 + 6*x^3 + x^4 + log(x)*(5*x^2 + 6*x^3 + x^4)) - log 
(x)*(x^3 + x^4) - log(x + 5)^2*(600*x + 100*x^2 + 500) + log(((x^2*log(x)) 
/25 + (log(x + 5)*(25*x + 100))/25)/(x*log(x + 5)))*(log(x + 5)^2*(1225*x 
+ 475*x^2 + 50*x^3 + 500) + log(x + 5)*log(x)*(5*x^2 + 11*x^3 + 2*x^4)))/( 
log(x + 5)^2*(225*x + 25*x^2 + 500) + log(x + 5)*log(x)*(5*x^2 + x^3)),x)
 

Output:

(log(((x^2*log(x))/25 + (log(x + 5)*(25*x + 100))/25)/(x*log(x + 5)))*(20* 
x^2 + 29*x^3 + 10*x^4 + x^5))/(x*(x + 4)*(x + 5))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.32 \[ \int \frac {\left (-x^3-x^4\right ) \log (x)+\left (5 x^2+6 x^3+x^4+\left (5 x^2+6 x^3+x^4\right ) \log (x)\right ) \log (5+x)+\left (-500-600 x-100 x^2\right ) \log ^2(5+x)+\left (\left (5 x^2+11 x^3+2 x^4\right ) \log (x) \log (5+x)+\left (500+1225 x+475 x^2+50 x^3\right ) \log ^2(5+x)\right ) \log \left (\frac {x^2 \log (x)+(100+25 x) \log (5+x)}{25 x \log (5+x)}\right )}{\left (5 x^2+x^3\right ) \log (x) \log (5+x)+\left (500+225 x+25 x^2\right ) \log ^2(5+x)} \, dx=\mathrm {log}\left (\frac {25 \,\mathrm {log}\left (x +5\right ) x +100 \,\mathrm {log}\left (x +5\right )+\mathrm {log}\left (x \right ) x^{2}}{25 \,\mathrm {log}\left (x +5\right ) x}\right ) x \left (x +1\right ) \] Input:

int((((50*x^3+475*x^2+1225*x+500)*log(5+x)^2+(2*x^4+11*x^3+5*x^2)*log(x)*l 
og(5+x))*log(1/25*((25*x+100)*log(5+x)+x^2*log(x))/x/log(5+x))+(-100*x^2-6 
00*x-500)*log(5+x)^2+((x^4+6*x^3+5*x^2)*log(x)+x^4+6*x^3+5*x^2)*log(5+x)+( 
-x^4-x^3)*log(x))/((25*x^2+225*x+500)*log(5+x)^2+(x^3+5*x^2)*log(x)*log(5+ 
x)),x)
 

Output:

log((25*log(x + 5)*x + 100*log(x + 5) + log(x)*x**2)/(25*log(x + 5)*x))*x* 
(x + 1)