\(\int \frac {(-20+6 x+(10-2 x+100 x^2-40 x^3+4 x^4) \log (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3})) \log (3 x \log (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}))}{(5 x-x^2+50 x^3-20 x^4+2 x^5) \log (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3})} \, dx\) [1120]

Optimal result

Integrand size = 134, antiderivative size = 19 \[ \int \frac {\left (-20+6 x+\left (10-2 x+100 x^2-40 x^3+4 x^4\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right ) \log \left (3 x \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right )}{\left (5 x-x^2+50 x^3-20 x^4+2 x^5\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )} \, dx=\log ^2\left (3 x \log \left (2-\frac {1}{(-5+x) x^2}\right )\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {\left (-20+6 x+\left (10-2 x+100 x^2-40 x^3+4 x^4\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right ) \log \left (3 x \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right )}{\left (5 x-x^2+50 x^3-20 x^4+2 x^5\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )} \, dx=\log ^2\left (3 x \log \left (\frac {-1-10 x^2+2 x^3}{(-5+x) x^2}\right )\right ) \] Input:

Integrate[((-20 + 6*x + (10 - 2*x + 100*x^2 - 40*x^3 + 4*x^4)*Log[(-1 - 10 
*x^2 + 2*x^3)/(-5*x^2 + x^3)])*Log[3*x*Log[(-1 - 10*x^2 + 2*x^3)/(-5*x^2 + 
 x^3)]])/((5*x - x^2 + 50*x^3 - 20*x^4 + 2*x^5)*Log[(-1 - 10*x^2 + 2*x^3)/ 
(-5*x^2 + x^3)]),x]
 

Output:

Log[3*x*Log[(-1 - 10*x^2 + 2*x^3)/((-5 + x)*x^2)]]^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.

\(\displaystyle \int \frac {\left (\left (4 x^4-40 x^3+100 x^2-2 x+10\right ) \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )+6 x-20\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )\right )}{\left (2 x^5-20 x^4+50 x^3-x^2+5 x\right ) \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )} \, dx\)

\(\Big \downarrow \) 2026

\(\displaystyle \int \frac {\left (\left (4 x^4-40 x^3+100 x^2-2 x+10\right ) \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )+6 x-20\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )\right )}{x \left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )}dx\)

\(\Big \downarrow \) 2463

\(\displaystyle \int \left (\frac {\left (\left (4 x^4-40 x^3+100 x^2-2 x+10\right ) \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )+6 x-20\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )\right )}{(5-x) x \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )}+\frac {2 x \left (\left (4 x^4-40 x^3+100 x^2-2 x+10\right ) \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )+6 x-20\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{x^3-5 x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {2 \left (\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+3 x-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 27

\(\displaystyle 2 \int -\frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {-2 x^3+10 x^2+1}{(5-x) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {-2 x^3+10 x^2+1}{(5-x) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {-2 x^3+10 x^2+1}{(5-x) x^2}\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {-2 x^3+10 x^2+1}{(5-x) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {-2 x^3+10 x^2+1}{(5-x) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {-2 x^3+10 x^2+1}{(5-x) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

\(\Big \downarrow \) 7239

\(\displaystyle -2 \int \frac {\left (-3 x-\left (2 x^4-20 x^3+50 x^2-x+5\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )+10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{(5-x) x \left (-2 x^3+10 x^2+1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}dx\)

\(\Big \downarrow \) 7293

\(\displaystyle -2 \int \left (\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 (x-5) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {\left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{5 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}-\frac {2 x \left (2 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^4-20 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^3+50 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x^2-\log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right ) x+3 x+5 \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )-10\right ) \log \left (3 x \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )\right )}{\left (2 x^3-10 x^2-1\right ) \log \left (\frac {2 x^3-10 x^2-1}{(x-5) x^2}\right )}\right )dx\)

Input:

Int[((-20 + 6*x + (10 - 2*x + 100*x^2 - 40*x^3 + 4*x^4)*Log[(-1 - 10*x^2 + 
 2*x^3)/(-5*x^2 + x^3)])*Log[3*x*Log[(-1 - 10*x^2 + 2*x^3)/(-5*x^2 + x^3)] 
])/((5*x - x^2 + 50*x^3 - 20*x^4 + 2*x^5)*Log[(-1 - 10*x^2 + 2*x^3)/(-5*x^ 
2 + x^3)]),x]
 

Output:

$Aborted
 

Maple [F]

\[\int \frac {\left (\left (4 x^{4}-40 x^{3}+100 x^{2}-2 x +10\right ) \ln \left (\frac {2 x^{3}-10 x^{2}-1}{x^{3}-5 x^{2}}\right )+6 x -20\right ) \ln \left (3 x \ln \left (\frac {2 x^{3}-10 x^{2}-1}{x^{3}-5 x^{2}}\right )\right )}{\left (2 x^{5}-20 x^{4}+50 x^{3}-x^{2}+5 x \right ) \ln \left (\frac {2 x^{3}-10 x^{2}-1}{x^{3}-5 x^{2}}\right )}d x\]

Input:

int(((4*x^4-40*x^3+100*x^2-2*x+10)*ln((2*x^3-10*x^2-1)/(x^3-5*x^2))+6*x-20 
)*ln(3*x*ln((2*x^3-10*x^2-1)/(x^3-5*x^2)))/(2*x^5-20*x^4+50*x^3-x^2+5*x)/l 
n((2*x^3-10*x^2-1)/(x^3-5*x^2)),x)
 

Output:

int(((4*x^4-40*x^3+100*x^2-2*x+10)*ln((2*x^3-10*x^2-1)/(x^3-5*x^2))+6*x-20 
)*ln(3*x*ln((2*x^3-10*x^2-1)/(x^3-5*x^2)))/(2*x^5-20*x^4+50*x^3-x^2+5*x)/l 
n((2*x^3-10*x^2-1)/(x^3-5*x^2)),x)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \frac {\left (-20+6 x+\left (10-2 x+100 x^2-40 x^3+4 x^4\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right ) \log \left (3 x \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right )}{\left (5 x-x^2+50 x^3-20 x^4+2 x^5\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )} \, dx=\log \left (3 \, x \log \left (\frac {2 \, x^{3} - 10 \, x^{2} - 1}{x^{3} - 5 \, x^{2}}\right )\right )^{2} \] Input:

integrate(((4*x^4-40*x^3+100*x^2-2*x+10)*log((2*x^3-10*x^2-1)/(x^3-5*x^2)) 
+6*x-20)*log(3*x*log((2*x^3-10*x^2-1)/(x^3-5*x^2)))/(2*x^5-20*x^4+50*x^3-x 
^2+5*x)/log((2*x^3-10*x^2-1)/(x^3-5*x^2)),x, algorithm="fricas")
 

Output:

log(3*x*log((2*x^3 - 10*x^2 - 1)/(x^3 - 5*x^2)))^2
 

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {\left (-20+6 x+\left (10-2 x+100 x^2-40 x^3+4 x^4\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right ) \log \left (3 x \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right )}{\left (5 x-x^2+50 x^3-20 x^4+2 x^5\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )} \, dx=\log {\left (3 x \log {\left (\frac {2 x^{3} - 10 x^{2} - 1}{x^{3} - 5 x^{2}} \right )} \right )}^{2} \] Input:

integrate(((4*x**4-40*x**3+100*x**2-2*x+10)*ln((2*x**3-10*x**2-1)/(x**3-5* 
x**2))+6*x-20)*ln(3*x*ln((2*x**3-10*x**2-1)/(x**3-5*x**2)))/(2*x**5-20*x** 
4+50*x**3-x**2+5*x)/ln((2*x**3-10*x**2-1)/(x**3-5*x**2)),x)
 

Output:

log(3*x*log((2*x**3 - 10*x**2 - 1)/(x**3 - 5*x**2)))**2
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 124 vs. \(2 (19) = 38\).

Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 6.53 \[ \int \frac {\left (-20+6 x+\left (10-2 x+100 x^2-40 x^3+4 x^4\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right ) \log \left (3 x \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right )}{\left (5 x-x^2+50 x^3-20 x^4+2 x^5\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )} \, dx=2 \, {\left (\log \left (x\right ) + \log \left (\log \left (2 \, x^{3} - 10 \, x^{2} - 1\right ) - \log \left (x - 5\right ) - 2 \, \log \left (x\right )\right )\right )} \log \left (3 \, x \log \left (\frac {2 \, x^{3} - 10 \, x^{2} - 1}{x^{3} - 5 \, x^{2}}\right )\right ) - \log \left (x\right )^{2} - 2 \, \log \left (x\right ) \log \left (\log \left (2 \, x^{3} - 10 \, x^{2} - 1\right ) - \log \left (x - 5\right ) - 2 \, \log \left (x\right )\right ) - \log \left (\log \left (2 \, x^{3} - 10 \, x^{2} - 1\right ) - \log \left (x - 5\right ) - 2 \, \log \left (x\right )\right )^{2} \] Input:

integrate(((4*x^4-40*x^3+100*x^2-2*x+10)*log((2*x^3-10*x^2-1)/(x^3-5*x^2)) 
+6*x-20)*log(3*x*log((2*x^3-10*x^2-1)/(x^3-5*x^2)))/(2*x^5-20*x^4+50*x^3-x 
^2+5*x)/log((2*x^3-10*x^2-1)/(x^3-5*x^2)),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

2*(log(x) + log(log(2*x^3 - 10*x^2 - 1) - log(x - 5) - 2*log(x)))*log(3*x* 
log((2*x^3 - 10*x^2 - 1)/(x^3 - 5*x^2))) - log(x)^2 - 2*log(x)*log(log(2*x 
^3 - 10*x^2 - 1) - log(x - 5) - 2*log(x)) - log(log(2*x^3 - 10*x^2 - 1) - 
log(x - 5) - 2*log(x))^2
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (19) = 38\).

Time = 0.56 (sec) , antiderivative size = 92, normalized size of antiderivative = 4.84 \[ \int \frac {\left (-20+6 x+\left (10-2 x+100 x^2-40 x^3+4 x^4\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right ) \log \left (3 x \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right )}{\left (5 x-x^2+50 x^3-20 x^4+2 x^5\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )} \, dx=\log \left (x\right )^{2} + 2 \, {\left (\log \left (x\right ) + \log \left (\log \left (2 \, x^{3} - 10 \, x^{2} - 1\right ) - \log \left (x - 5\right ) - 2 \, \log \left (x\right )\right )\right )} \log \left (3 \, \log \left (\frac {2 \, x^{3} - 10 \, x^{2} - 1}{x^{3} - 5 \, x^{2}}\right )\right ) - \log \left (\log \left (2 \, x^{3} - 10 \, x^{2} - 1\right ) - \log \left (x - 5\right ) - 2 \, \log \left (x\right )\right )^{2} \] Input:

integrate(((4*x^4-40*x^3+100*x^2-2*x+10)*log((2*x^3-10*x^2-1)/(x^3-5*x^2)) 
+6*x-20)*log(3*x*log((2*x^3-10*x^2-1)/(x^3-5*x^2)))/(2*x^5-20*x^4+50*x^3-x 
^2+5*x)/log((2*x^3-10*x^2-1)/(x^3-5*x^2)),x, algorithm="giac")
 

Output:

log(x)^2 + 2*(log(x) + log(log(2*x^3 - 10*x^2 - 1) - log(x - 5) - 2*log(x) 
))*log(3*log((2*x^3 - 10*x^2 - 1)/(x^3 - 5*x^2))) - log(log(2*x^3 - 10*x^2 
 - 1) - log(x - 5) - 2*log(x))^2
 

Mupad [B] (verification not implemented)

Time = 3.69 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.74 \[ \int \frac {\left (-20+6 x+\left (10-2 x+100 x^2-40 x^3+4 x^4\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right ) \log \left (3 x \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right )}{\left (5 x-x^2+50 x^3-20 x^4+2 x^5\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )} \, dx={\ln \left (3\,x\,\ln \left (\frac {-2\,x^3+10\,x^2+1}{5\,x^2-x^3}\right )\right )}^2 \] Input:

int((log(3*x*log((10*x^2 - 2*x^3 + 1)/(5*x^2 - x^3)))*(6*x + log((10*x^2 - 
 2*x^3 + 1)/(5*x^2 - x^3))*(100*x^2 - 2*x - 40*x^3 + 4*x^4 + 10) - 20))/(l 
og((10*x^2 - 2*x^3 + 1)/(5*x^2 - x^3))*(5*x - x^2 + 50*x^3 - 20*x^4 + 2*x^ 
5)),x)
 

Output:

log(3*x*log((10*x^2 - 2*x^3 + 1)/(5*x^2 - x^3)))^2
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \frac {\left (-20+6 x+\left (10-2 x+100 x^2-40 x^3+4 x^4\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right ) \log \left (3 x \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )\right )}{\left (5 x-x^2+50 x^3-20 x^4+2 x^5\right ) \log \left (\frac {-1-10 x^2+2 x^3}{-5 x^2+x^3}\right )} \, dx={\mathrm {log}\left (3 \,\mathrm {log}\left (\frac {2 x^{3}-10 x^{2}-1}{x^{3}-5 x^{2}}\right ) x \right )}^{2} \] Input:

int(((4*x^4-40*x^3+100*x^2-2*x+10)*log((2*x^3-10*x^2-1)/(x^3-5*x^2))+6*x-2 
0)*log(3*x*log((2*x^3-10*x^2-1)/(x^3-5*x^2)))/(2*x^5-20*x^4+50*x^3-x^2+5*x 
)/log((2*x^3-10*x^2-1)/(x^3-5*x^2)),x)
 

Output:

log(3*log((2*x**3 - 10*x**2 - 1)/(x**3 - 5*x**2))*x)**2