\(\int \frac {(-75-53 x-2 x^2) \log ^2(25+x)+(-15 x-5 x^2) \log (\frac {1}{3} (-3 x-x^2))+(-1125-420 x-15 x^2) \log (25+x) \log (\frac {1}{3} (-3 x-x^2))+(225+84 x+3 x^2) \log ^2(25+x) \log (\frac {1}{3} (-3 x-x^2)) \log (\log (\frac {1}{3} (-3 x-x^2)))+((375+140 x+5 x^2) \log (25+x) \log (\frac {1}{3} (-3 x-x^2))+(-75-28 x-x^2) \log ^2(25+x) \log (\frac {1}{3} (-3 x-x^2)) \log (\log (\frac {1}{3} (-3 x-x^2)))) \log (\frac {5-\log (25+x) \log (\log (\frac {1}{3} (-3 x-x^2)))}{\log (25+x)})}{(-375-140 x-5 x^2) \log (25+x) \log (\frac {1}{3} (-3 x-x^2))+(75+28 x+x^2) \log ^2(25+x) \log (\frac {1}{3} (-3 x-x^2)) \log (\log (\frac {1}{3} (-3 x-x^2)))} \, dx\) [870]

Optimal result

Integrand size = 302, antiderivative size = 30 \[ \int \frac {\left (-75-53 x-2 x^2\right ) \log ^2(25+x)+\left (-15 x-5 x^2\right ) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-1125-420 x-15 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (225+84 x+3 x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )+\left (\left (375+140 x+5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-75-28 x-x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )\right ) \log \left (\frac {5-\log (25+x) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )}{\log (25+x)}\right )}{\left (-375-140 x-5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (75+28 x+x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )} \, dx=x \left (3-\log \left (\frac {5}{\log (25+x)}-\log \left (\log \left (\frac {1}{3} (-3-x) x\right )\right )\right )\right ) \] Output:

(3-ln(5/ln(x+25)-ln(ln(1/3*x*(-3-x)))))*x
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-75-53 x-2 x^2\right ) \log ^2(25+x)+\left (-15 x-5 x^2\right ) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-1125-420 x-15 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (225+84 x+3 x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )+\left (\left (375+140 x+5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-75-28 x-x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )\right ) \log \left (\frac {5-\log (25+x) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )}{\log (25+x)}\right )}{\left (-375-140 x-5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (75+28 x+x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )} \, dx=3 x-x \log \left (\frac {5}{\log (25+x)}-\log \left (\log \left (-\frac {1}{3} x (3+x)\right )\right )\right ) \] Input:

Integrate[((-75 - 53*x - 2*x^2)*Log[25 + x]^2 + (-15*x - 5*x^2)*Log[(-3*x 
- x^2)/3] + (-1125 - 420*x - 15*x^2)*Log[25 + x]*Log[(-3*x - x^2)/3] + (22 
5 + 84*x + 3*x^2)*Log[25 + x]^2*Log[(-3*x - x^2)/3]*Log[Log[(-3*x - x^2)/3 
]] + ((375 + 140*x + 5*x^2)*Log[25 + x]*Log[(-3*x - x^2)/3] + (-75 - 28*x 
- x^2)*Log[25 + x]^2*Log[(-3*x - x^2)/3]*Log[Log[(-3*x - x^2)/3]])*Log[(5 
- Log[25 + x]*Log[Log[(-3*x - x^2)/3]])/Log[25 + x]])/((-375 - 140*x - 5*x 
^2)*Log[25 + x]*Log[(-3*x - x^2)/3] + (75 + 28*x + x^2)*Log[25 + x]^2*Log[ 
(-3*x - x^2)/3]*Log[Log[(-3*x - x^2)/3]]),x]
 

Output:

3*x - x*Log[5/Log[25 + x] - Log[Log[-1/3*(x*(3 + 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.

\(\displaystyle \int \frac {\left (-2 x^2-53 x-75\right ) \log ^2(x+25)+\left (3 x^2+84 x+225\right ) \log \left (\frac {1}{3} \left (-x^2-3 x\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-x^2-3 x\right )\right )\right ) \log ^2(x+25)+\left (\left (-x^2-28 x-75\right ) \log \left (\frac {1}{3} \left (-x^2-3 x\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-x^2-3 x\right )\right )\right ) \log ^2(x+25)+\left (5 x^2+140 x+375\right ) \log \left (\frac {1}{3} \left (-x^2-3 x\right )\right ) \log (x+25)\right ) \log \left (\frac {5-\log (x+25) \log \left (\log \left (\frac {1}{3} \left (-x^2-3 x\right )\right )\right )}{\log (x+25)}\right )+\left (-15 x^2-420 x-1125\right ) \log \left (\frac {1}{3} \left (-x^2-3 x\right )\right ) \log (x+25)+\left (-5 x^2-15 x\right ) \log \left (\frac {1}{3} \left (-x^2-3 x\right )\right )}{\left (x^2+28 x+75\right ) \log \left (\frac {1}{3} \left (-x^2-3 x\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-x^2-3 x\right )\right )\right ) \log ^2(x+25)+\left (-5 x^2-140 x-375\right ) \log \left (\frac {1}{3} \left (-x^2-3 x\right )\right ) \log (x+25)} \, dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \left (-\frac {5 x}{(x+25) \log (x+25) \left (\log (x+25) \log \left (\log \left (-\frac {1}{3} x (x+3)\right )\right )-5\right )}-\log \left (\frac {5}{\log (x+25)}-\log \left (\log \left (-\frac {1}{3} x (x+3)\right )\right )\right )+\frac {\log (x+25) \left (-2 x+3 (x+3) \log \left (-\frac {1}{3} x (x+3)\right ) \log \left (\log \left (-\frac {1}{3} x (x+3)\right )\right )-3\right )}{(x+3) \log \left (-\frac {1}{3} x (x+3)\right ) \left (\log (x+25) \log \left (\log \left (-\frac {1}{3} x (x+3)\right )\right )-5\right )}-\frac {15}{\log (x+25) \log \left (\log \left (-\frac {1}{3} x (x+3)\right )\right )-5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -5 \int \frac {1}{\log (x+25) \left (\log (x+25) \log \left (\log \left (-\frac {1}{3} x (x+3)\right )\right )-5\right )}dx+125 \int \frac {1}{(x+25) \log (x+25) \left (\log (x+25) \log \left (\log \left (-\frac {1}{3} x (x+3)\right )\right )-5\right )}dx-2 \int \frac {\log (x+25)}{\log \left (-\frac {1}{3} x (x+3)\right ) \left (\log (x+25) \log \left (\log \left (-\frac {1}{3} x (x+3)\right )\right )-5\right )}dx+3 \int \frac {\log (x+25)}{(x+3) \log \left (-\frac {1}{3} x (x+3)\right ) \left (\log (x+25) \log \left (\log \left (-\frac {1}{3} x (x+3)\right )\right )-5\right )}dx-\int \log \left (\frac {5}{\log (x+25)}-\log \left (\log \left (-\frac {1}{3} x (x+3)\right )\right )\right )dx+3 x\)

Input:

Int[((-75 - 53*x - 2*x^2)*Log[25 + x]^2 + (-15*x - 5*x^2)*Log[(-3*x - x^2) 
/3] + (-1125 - 420*x - 15*x^2)*Log[25 + x]*Log[(-3*x - x^2)/3] + (225 + 84 
*x + 3*x^2)*Log[25 + x]^2*Log[(-3*x - x^2)/3]*Log[Log[(-3*x - x^2)/3]] + ( 
(375 + 140*x + 5*x^2)*Log[25 + x]*Log[(-3*x - x^2)/3] + (-75 - 28*x - x^2) 
*Log[25 + x]^2*Log[(-3*x - x^2)/3]*Log[Log[(-3*x - x^2)/3]])*Log[(5 - Log[ 
25 + x]*Log[Log[(-3*x - x^2)/3]])/Log[25 + x]])/((-375 - 140*x - 5*x^2)*Lo 
g[25 + x]*Log[(-3*x - x^2)/3] + (75 + 28*x + x^2)*Log[25 + x]^2*Log[(-3*x 
- x^2)/3]*Log[Log[(-3*x - x^2)/3]]),x]
 

Output:

$Aborted
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.02 (sec) , antiderivative size = 876, normalized size of antiderivative = 29.20

\[\text {Expression too large to display}\]

Input:

int((((-x^2-28*x-75)*ln(-1/3*x^2-x)*ln(x+25)^2*ln(ln(-1/3*x^2-x))+(5*x^2+1 
40*x+375)*ln(-1/3*x^2-x)*ln(x+25))*ln((-ln(x+25)*ln(ln(-1/3*x^2-x))+5)/ln( 
x+25))+(3*x^2+84*x+225)*ln(-1/3*x^2-x)*ln(x+25)^2*ln(ln(-1/3*x^2-x))+(-2*x 
^2-53*x-75)*ln(x+25)^2+(-15*x^2-420*x-1125)*ln(-1/3*x^2-x)*ln(x+25)+(-5*x^ 
2-15*x)*ln(-1/3*x^2-x))/((x^2+28*x+75)*ln(-1/3*x^2-x)*ln(x+25)^2*ln(ln(-1/ 
3*x^2-x))+(-5*x^2-140*x-375)*ln(-1/3*x^2-x)*ln(x+25)),x)
 

Output:

-x*ln(ln(x+25)*ln(-ln(3)+I*Pi+ln(x)+ln(3+x)-1/2*I*Pi*csgn(I*(3+x)*x)*(-csg 
n(I*(3+x)*x)+csgn(I*x))*(-csgn(I*(3+x)*x)+csgn(I*(3+x)))+I*Pi*csgn(I*(3+x) 
*x)^2*(csgn(I*(3+x)*x)-1))-5)+x*ln(ln(x+25))+1/2*I*Pi*x*csgn(I*(ln(x+25)*l 
n(-ln(3)+I*Pi+ln(x)+ln(3+x)+1/2*I*Pi*csgn(I*(3+x)*x)*(csgn(I*(3+x)*x)-csgn 
(I*x))*(-csgn(I*(3+x)*x)+csgn(I*(3+x)))+I*Pi*csgn(I*(3+x)*x)^2*(csgn(I*(3+ 
x)*x)-1))-5))*csgn(I/ln(x+25))*csgn(I*(ln(x+25)*ln(-ln(3)+I*Pi+ln(x)+ln(3+ 
x)+1/2*I*Pi*csgn(I*(3+x)*x)*(csgn(I*(3+x)*x)-csgn(I*x))*(-csgn(I*(3+x)*x)+ 
csgn(I*(3+x)))+I*Pi*csgn(I*(3+x)*x)^2*(csgn(I*(3+x)*x)-1))-5)/ln(x+25))-1/ 
2*I*Pi*x*csgn(I*(ln(x+25)*ln(-ln(3)+I*Pi+ln(x)+ln(3+x)+1/2*I*Pi*csgn(I*(3+ 
x)*x)*(csgn(I*(3+x)*x)-csgn(I*x))*(-csgn(I*(3+x)*x)+csgn(I*(3+x)))+I*Pi*cs 
gn(I*(3+x)*x)^2*(csgn(I*(3+x)*x)-1))-5))*csgn(I*(ln(x+25)*ln(-ln(3)+I*Pi+l 
n(x)+ln(3+x)+1/2*I*Pi*csgn(I*(3+x)*x)*(csgn(I*(3+x)*x)-csgn(I*x))*(-csgn(I 
*(3+x)*x)+csgn(I*(3+x)))+I*Pi*csgn(I*(3+x)*x)^2*(csgn(I*(3+x)*x)-1))-5)/ln 
(x+25))^2-1/2*I*Pi*x*csgn(I/ln(x+25))*csgn(I*(ln(x+25)*ln(-ln(3)+I*Pi+ln(x 
)+ln(3+x)+1/2*I*Pi*csgn(I*(3+x)*x)*(csgn(I*(3+x)*x)-csgn(I*x))*(-csgn(I*(3 
+x)*x)+csgn(I*(3+x)))+I*Pi*csgn(I*(3+x)*x)^2*(csgn(I*(3+x)*x)-1))-5)/ln(x+ 
25))^2-1/2*I*Pi*x*csgn(I*(ln(x+25)*ln(-ln(3)+I*Pi+ln(x)+ln(3+x)+1/2*I*Pi*c 
sgn(I*(3+x)*x)*(csgn(I*(3+x)*x)-csgn(I*x))*(-csgn(I*(3+x)*x)+csgn(I*(3+x)) 
)+I*Pi*csgn(I*(3+x)*x)^2*(csgn(I*(3+x)*x)-1))-5)/ln(x+25))^3+I*Pi*x*csgn(I 
*(ln(x+25)*ln(-ln(3)+I*Pi+ln(x)+ln(3+x)+1/2*I*Pi*csgn(I*(3+x)*x)*(csgn(...
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {\left (-75-53 x-2 x^2\right ) \log ^2(25+x)+\left (-15 x-5 x^2\right ) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-1125-420 x-15 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (225+84 x+3 x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )+\left (\left (375+140 x+5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-75-28 x-x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )\right ) \log \left (\frac {5-\log (25+x) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )}{\log (25+x)}\right )}{\left (-375-140 x-5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (75+28 x+x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )} \, dx=-x \log \left (-\frac {\log \left (x + 25\right ) \log \left (\log \left (-\frac {1}{3} \, x^{2} - x\right )\right ) - 5}{\log \left (x + 25\right )}\right ) + 3 \, x \] Input:

integrate((((-x^2-28*x-75)*log(-1/3*x^2-x)*log(x+25)^2*log(log(-1/3*x^2-x) 
)+(5*x^2+140*x+375)*log(-1/3*x^2-x)*log(x+25))*log((-log(x+25)*log(log(-1/ 
3*x^2-x))+5)/log(x+25))+(3*x^2+84*x+225)*log(-1/3*x^2-x)*log(x+25)^2*log(l 
og(-1/3*x^2-x))+(-2*x^2-53*x-75)*log(x+25)^2+(-15*x^2-420*x-1125)*log(-1/3 
*x^2-x)*log(x+25)+(-5*x^2-15*x)*log(-1/3*x^2-x))/((x^2+28*x+75)*log(-1/3*x 
^2-x)*log(x+25)^2*log(log(-1/3*x^2-x))+(-5*x^2-140*x-375)*log(-1/3*x^2-x)* 
log(x+25)),x, algorithm="fricas")
 

Output:

-x*log(-(log(x + 25)*log(log(-1/3*x^2 - x)) - 5)/log(x + 25)) + 3*x
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-75-53 x-2 x^2\right ) \log ^2(25+x)+\left (-15 x-5 x^2\right ) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-1125-420 x-15 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (225+84 x+3 x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )+\left (\left (375+140 x+5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-75-28 x-x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )\right ) \log \left (\frac {5-\log (25+x) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )}{\log (25+x)}\right )}{\left (-375-140 x-5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (75+28 x+x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )} \, dx=\text {Timed out} \] Input:

integrate((((-x**2-28*x-75)*ln(-1/3*x**2-x)*ln(x+25)**2*ln(ln(-1/3*x**2-x) 
)+(5*x**2+140*x+375)*ln(-1/3*x**2-x)*ln(x+25))*ln((-ln(x+25)*ln(ln(-1/3*x* 
*2-x))+5)/ln(x+25))+(3*x**2+84*x+225)*ln(-1/3*x**2-x)*ln(x+25)**2*ln(ln(-1 
/3*x**2-x))+(-2*x**2-53*x-75)*ln(x+25)**2+(-15*x**2-420*x-1125)*ln(-1/3*x* 
*2-x)*ln(x+25)+(-5*x**2-15*x)*ln(-1/3*x**2-x))/((x**2+28*x+75)*ln(-1/3*x** 
2-x)*ln(x+25)**2*ln(ln(-1/3*x**2-x))+(-5*x**2-140*x-375)*ln(-1/3*x**2-x)*l 
n(x+25)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-75-53 x-2 x^2\right ) \log ^2(25+x)+\left (-15 x-5 x^2\right ) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-1125-420 x-15 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (225+84 x+3 x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )+\left (\left (375+140 x+5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-75-28 x-x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )\right ) \log \left (\frac {5-\log (25+x) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )}{\log (25+x)}\right )}{\left (-375-140 x-5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (75+28 x+x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )} \, dx=-x \log \left (-\log \left (x + 25\right ) \log \left (-\log \left (3\right ) + \log \left (x\right ) + \log \left (-x - 3\right )\right ) + 5\right ) + x \log \left (\log \left (x + 25\right )\right ) + 3 \, x \] Input:

integrate((((-x^2-28*x-75)*log(-1/3*x^2-x)*log(x+25)^2*log(log(-1/3*x^2-x) 
)+(5*x^2+140*x+375)*log(-1/3*x^2-x)*log(x+25))*log((-log(x+25)*log(log(-1/ 
3*x^2-x))+5)/log(x+25))+(3*x^2+84*x+225)*log(-1/3*x^2-x)*log(x+25)^2*log(l 
og(-1/3*x^2-x))+(-2*x^2-53*x-75)*log(x+25)^2+(-15*x^2-420*x-1125)*log(-1/3 
*x^2-x)*log(x+25)+(-5*x^2-15*x)*log(-1/3*x^2-x))/((x^2+28*x+75)*log(-1/3*x 
^2-x)*log(x+25)^2*log(log(-1/3*x^2-x))+(-5*x^2-140*x-375)*log(-1/3*x^2-x)* 
log(x+25)),x, algorithm="maxima")
 

Output:

-x*log(-log(x + 25)*log(-log(3) + log(x) + log(-x - 3)) + 5) + x*log(log(x 
 + 25)) + 3*x
 

Giac [A] (verification not implemented)

Time = 0.87 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {\left (-75-53 x-2 x^2\right ) \log ^2(25+x)+\left (-15 x-5 x^2\right ) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-1125-420 x-15 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (225+84 x+3 x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )+\left (\left (375+140 x+5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-75-28 x-x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )\right ) \log \left (\frac {5-\log (25+x) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )}{\log (25+x)}\right )}{\left (-375-140 x-5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (75+28 x+x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )} \, dx=-x \log \left (-\log \left (x + 25\right ) \log \left (\log \left (-\frac {1}{3} \, x^{2} - x\right )\right ) + 5\right ) + x \log \left (\log \left (x + 25\right )\right ) + 3 \, x \] Input:

integrate((((-x^2-28*x-75)*log(-1/3*x^2-x)*log(x+25)^2*log(log(-1/3*x^2-x) 
)+(5*x^2+140*x+375)*log(-1/3*x^2-x)*log(x+25))*log((-log(x+25)*log(log(-1/ 
3*x^2-x))+5)/log(x+25))+(3*x^2+84*x+225)*log(-1/3*x^2-x)*log(x+25)^2*log(l 
og(-1/3*x^2-x))+(-2*x^2-53*x-75)*log(x+25)^2+(-15*x^2-420*x-1125)*log(-1/3 
*x^2-x)*log(x+25)+(-5*x^2-15*x)*log(-1/3*x^2-x))/((x^2+28*x+75)*log(-1/3*x 
^2-x)*log(x+25)^2*log(log(-1/3*x^2-x))+(-5*x^2-140*x-375)*log(-1/3*x^2-x)* 
log(x+25)),x, algorithm="giac")
 

Output:

-x*log(-log(x + 25)*log(log(-1/3*x^2 - x)) + 5) + x*log(log(x + 25)) + 3*x
 

Mupad [B] (verification not implemented)

Time = 8.55 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {\left (-75-53 x-2 x^2\right ) \log ^2(25+x)+\left (-15 x-5 x^2\right ) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-1125-420 x-15 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (225+84 x+3 x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )+\left (\left (375+140 x+5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-75-28 x-x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )\right ) \log \left (\frac {5-\log (25+x) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )}{\log (25+x)}\right )}{\left (-375-140 x-5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (75+28 x+x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )} \, dx=-x\,\left (\ln \left (-\frac {\ln \left (x+25\right )\,\ln \left (\ln \left (-\frac {x^2}{3}-x\right )\right )-5}{\ln \left (x+25\right )}\right )-3\right ) \] Input:

int((log(x + 25)^2*(53*x + 2*x^2 + 75) - log(-(log(x + 25)*log(log(- x - x 
^2/3)) - 5)/log(x + 25))*(log(x + 25)*log(- x - x^2/3)*(140*x + 5*x^2 + 37 
5) - log(x + 25)^2*log(log(- x - x^2/3))*log(- x - x^2/3)*(28*x + x^2 + 75 
)) + log(- x - x^2/3)*(15*x + 5*x^2) + log(x + 25)*log(- x - x^2/3)*(420*x 
 + 15*x^2 + 1125) - log(x + 25)^2*log(log(- x - x^2/3))*log(- x - x^2/3)*( 
84*x + 3*x^2 + 225))/(log(x + 25)*log(- x - x^2/3)*(140*x + 5*x^2 + 375) - 
 log(x + 25)^2*log(log(- x - x^2/3))*log(- x - x^2/3)*(28*x + x^2 + 75)),x 
)
                                                                                    
                                                                                    
 

Output:

-x*(log(-(log(x + 25)*log(log(- x - x^2/3)) - 5)/log(x + 25)) - 3)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.03 \[ \int \frac {\left (-75-53 x-2 x^2\right ) \log ^2(25+x)+\left (-15 x-5 x^2\right ) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-1125-420 x-15 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (225+84 x+3 x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )+\left (\left (375+140 x+5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (-75-28 x-x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )\right ) \log \left (\frac {5-\log (25+x) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )}{\log (25+x)}\right )}{\left (-375-140 x-5 x^2\right ) \log (25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )+\left (75+28 x+x^2\right ) \log ^2(25+x) \log \left (\frac {1}{3} \left (-3 x-x^2\right )\right ) \log \left (\log \left (\frac {1}{3} \left (-3 x-x^2\right )\right )\right )} \, dx=\frac {53 \,\mathrm {log}\left (\mathrm {log}\left (x +25\right )\right )}{2}-\frac {53 \,\mathrm {log}\left (\mathrm {log}\left (\mathrm {log}\left (-\frac {1}{3} x^{2}-x \right )\right ) \mathrm {log}\left (x +25\right )-5\right )}{2}-\mathrm {log}\left (\frac {-\mathrm {log}\left (\mathrm {log}\left (-\frac {1}{3} x^{2}-x \right )\right ) \mathrm {log}\left (x +25\right )+5}{\mathrm {log}\left (x +25\right )}\right ) x +\frac {53 \,\mathrm {log}\left (\frac {-\mathrm {log}\left (\mathrm {log}\left (-\frac {1}{3} x^{2}-x \right )\right ) \mathrm {log}\left (x +25\right )+5}{\mathrm {log}\left (x +25\right )}\right )}{2}+3 x \] Input:

int((((-x^2-28*x-75)*log(-1/3*x^2-x)*log(x+25)^2*log(log(-1/3*x^2-x))+(5*x 
^2+140*x+375)*log(-1/3*x^2-x)*log(x+25))*log((-log(x+25)*log(log(-1/3*x^2- 
x))+5)/log(x+25))+(3*x^2+84*x+225)*log(-1/3*x^2-x)*log(x+25)^2*log(log(-1/ 
3*x^2-x))+(-2*x^2-53*x-75)*log(x+25)^2+(-15*x^2-420*x-1125)*log(-1/3*x^2-x 
)*log(x+25)+(-5*x^2-15*x)*log(-1/3*x^2-x))/((x^2+28*x+75)*log(-1/3*x^2-x)* 
log(x+25)^2*log(log(-1/3*x^2-x))+(-5*x^2-140*x-375)*log(-1/3*x^2-x)*log(x+ 
25)),x)
 

Output:

(53*log(log(x + 25)) - 53*log(log(log(( - x**2 - 3*x)/3))*log(x + 25) - 5) 
 - 2*log(( - log(log(( - x**2 - 3*x)/3))*log(x + 25) + 5)/log(x + 25))*x + 
 53*log(( - log(log(( - x**2 - 3*x)/3))*log(x + 25) + 5)/log(x + 25)) + 6* 
x)/2