\(\int \frac {30 x+10 x^2+(384+544 x+264 x^2+54 x^3+4 x^4) \log (3 x+x^2)}{-60 x-35 x^2-5 x^3+(192 x+208 x^2+84 x^3+15 x^4+x^5) \log ^2(3 x+x^2)} \, dx\) [1429]

Optimal result

Integrand size = 88, antiderivative size = 23 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=2+\log \left (\frac {5}{(4+x)^2}-\log ^2\left (3 x+x^2\right )\right ) \] Output:

2+ln(5/(4+x)^2-ln(x^2+3*x)^2)
 

Mathematica [A] (verified)

Time = 0.46 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=-2 \log (4+x)+\log \left (5-16 \log ^2(x (3+x))-8 x \log ^2(x (3+x))-x^2 \log ^2(x (3+x))\right ) \] Input:

Integrate[(30*x + 10*x^2 + (384 + 544*x + 264*x^2 + 54*x^3 + 4*x^4)*Log[3* 
x + x^2])/(-60*x - 35*x^2 - 5*x^3 + (192*x + 208*x^2 + 84*x^3 + 15*x^4 + x 
^5)*Log[3*x + x^2]^2),x]
 

Output:

-2*Log[4 + x] + Log[5 - 16*Log[x*(3 + x)]^2 - 8*x*Log[x*(3 + x)]^2 - x^2*L 
og[x*(3 + 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.

\(\displaystyle \int \frac {10 x^2+\left (4 x^4+54 x^3+264 x^2+544 x+384\right ) \log \left (x^2+3 x\right )+30 x}{-5 x^3-35 x^2+\left (x^5+15 x^4+84 x^3+208 x^2+192 x\right ) \log ^2\left (x^2+3 x\right )-60 x} \, dx\)

\(\Big \downarrow \) 7292

\(\displaystyle \int \frac {-10 x^2-\left (4 x^4+54 x^3+264 x^2+544 x+384\right ) \log \left (x^2+3 x\right )-30 x}{x \left (x^2+7 x+12\right ) \left (-x^2 \log ^2(x (x+3))-8 x \log ^2(x (x+3))-16 \log ^2(x (x+3))+5\right )}dx\)

\(\Big \downarrow \) 7239

\(\displaystyle \int \frac {-10 x (x+3)-2 (2 x+3) (x+4)^3 \log (x (x+3))}{x \left (x^2+7 x+12\right ) \left (5-(x+4)^2 \log ^2(x (x+3))\right )}dx\)

\(\Big \downarrow \) 7279

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{6 x \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}-\frac {2 \left (2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))\right )}{3 (x+3) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}+\frac {2 x^4 \log (x (x+3))+27 x^3 \log (x (x+3))+5 x^2+132 x^2 \log (x (x+3))+15 x+272 x \log (x (x+3))+192 \log (x (x+3))}{2 (x+4) \left (x^2 \log ^2(x (x+3))+8 x \log ^2(x (x+3))+16 \log ^2(x (x+3))-5\right )}\right )dx\)

\(\Big \downarrow \) 7239

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

Input:

Int[(30*x + 10*x^2 + (384 + 544*x + 264*x^2 + 54*x^3 + 4*x^4)*Log[3*x + x^ 
2])/(-60*x - 35*x^2 - 5*x^3 + (192*x + 208*x^2 + 84*x^3 + 15*x^4 + x^5)*Lo 
g[3*x + x^2]^2),x]
 

Output:

$Aborted
 

Maple [A] (verified)

Time = 0.41 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09

method	result	size
risch	\(\ln \left (\ln \left (x^{2}+3 x \right )^{2}-\frac {5}{x^{2}+8 x +16}\right )\)	\(25\)
norman	\(-2 \ln \left (4+x \right )+\ln \left (\ln \left (x^{2}+3 x \right )^{2} x^{2}+8 \ln \left (x^{2}+3 x \right )^{2} x +16 \ln \left (x^{2}+3 x \right )^{2}-5\right )\)	\(50\)
parallelrisch	\(-2 \ln \left (4+x \right )+\ln \left (\ln \left (x^{2}+3 x \right )^{2} x^{2}+8 \ln \left (x^{2}+3 x \right )^{2} x +16 \ln \left (x^{2}+3 x \right )^{2}-5\right )\)	\(50\)

Input:

int(((4*x^4+54*x^3+264*x^2+544*x+384)*ln(x^2+3*x)+10*x^2+30*x)/((x^5+15*x^ 
4+84*x^3+208*x^2+192*x)*ln(x^2+3*x)^2-5*x^3-35*x^2-60*x),x,method=_RETURNV 
ERBOSE)
 

Output:

ln(ln(x^2+3*x)^2-5/(x^2+8*x+16))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=\log \left (\frac {{\left (x^{2} + 8 \, x + 16\right )} \log \left (x^{2} + 3 \, x\right )^{2} - 5}{x^{2} + 8 \, x + 16}\right ) \] Input:

integrate(((4*x^4+54*x^3+264*x^2+544*x+384)*log(x^2+3*x)+10*x^2+30*x)/((x^ 
5+15*x^4+84*x^3+208*x^2+192*x)*log(x^2+3*x)^2-5*x^3-35*x^2-60*x),x, algori 
thm="fricas")
 

Output:

log(((x^2 + 8*x + 16)*log(x^2 + 3*x)^2 - 5)/(x^2 + 8*x + 16))
 

Sympy [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=\log {\left (\log {\left (x^{2} + 3 x \right )}^{2} - \frac {5}{x^{2} + 8 x + 16} \right )} \] Input:

integrate(((4*x**4+54*x**3+264*x**2+544*x+384)*ln(x**2+3*x)+10*x**2+30*x)/ 
((x**5+15*x**4+84*x**3+208*x**2+192*x)*ln(x**2+3*x)**2-5*x**3-35*x**2-60*x 
),x)
 

Output:

log(log(x**2 + 3*x)**2 - 5/(x**2 + 8*x + 16))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (23) = 46\).

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=\log \left (\frac {{\left (x^{2} + 8 \, x + 16\right )} \log \left (x + 3\right )^{2} + 2 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (x + 3\right ) \log \left (x\right ) + {\left (x^{2} + 8 \, x + 16\right )} \log \left (x\right )^{2} - 5}{x^{2} + 8 \, x + 16}\right ) \] Input:

integrate(((4*x^4+54*x^3+264*x^2+544*x+384)*log(x^2+3*x)+10*x^2+30*x)/((x^ 
5+15*x^4+84*x^3+208*x^2+192*x)*log(x^2+3*x)^2-5*x^3-35*x^2-60*x),x, algori 
thm="maxima")
 

Output:

log(((x^2 + 8*x + 16)*log(x + 3)^2 + 2*(x^2 + 8*x + 16)*log(x + 3)*log(x) 
+ (x^2 + 8*x + 16)*log(x)^2 - 5)/(x^2 + 8*x + 16))
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).

Time = 0.17 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=\log \left (x^{2} \log \left (x^{2} + 3 \, x\right )^{2} + 8 \, x \log \left (x^{2} + 3 \, x\right )^{2} + 16 \, \log \left (x^{2} + 3 \, x\right )^{2} - 5\right ) - 2 \, \log \left (x + 4\right ) \] Input:

integrate(((4*x^4+54*x^3+264*x^2+544*x+384)*log(x^2+3*x)+10*x^2+30*x)/((x^ 
5+15*x^4+84*x^3+208*x^2+192*x)*log(x^2+3*x)^2-5*x^3-35*x^2-60*x),x, algori 
thm="giac")
 

Output:

log(x^2*log(x^2 + 3*x)^2 + 8*x*log(x^2 + 3*x)^2 + 16*log(x^2 + 3*x)^2 - 5) 
 - 2*log(x + 4)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=-\int \frac {30\,x+\ln \left (x^2+3\,x\right )\,\left (4\,x^4+54\,x^3+264\,x^2+544\,x+384\right )+10\,x^2}{60\,x+35\,x^2+5\,x^3-{\ln \left (x^2+3\,x\right )}^2\,\left (x^5+15\,x^4+84\,x^3+208\,x^2+192\,x\right )} \,d x \] Input:

int(-(30*x + log(3*x + x^2)*(544*x + 264*x^2 + 54*x^3 + 4*x^4 + 384) + 10* 
x^2)/(60*x + 35*x^2 + 5*x^3 - log(3*x + x^2)^2*(192*x + 208*x^2 + 84*x^3 + 
 15*x^4 + x^5)),x)
 

Output:

-int((30*x + log(3*x + x^2)*(544*x + 264*x^2 + 54*x^3 + 4*x^4 + 384) + 10* 
x^2)/(60*x + 35*x^2 + 5*x^3 - log(3*x + x^2)^2*(192*x + 208*x^2 + 84*x^3 + 
 15*x^4 + x^5)), x)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.48 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=\mathrm {log}\left (-\sqrt {5}+\mathrm {log}\left (x^{2}+3 x \right ) x +4 \,\mathrm {log}\left (x^{2}+3 x \right )\right )+\mathrm {log}\left (\sqrt {5}+\mathrm {log}\left (x^{2}+3 x \right ) x +4 \,\mathrm {log}\left (x^{2}+3 x \right )\right )-2 \,\mathrm {log}\left (x +4\right ) \] Input:

int(((4*x^4+54*x^3+264*x^2+544*x+384)*log(x^2+3*x)+10*x^2+30*x)/((x^5+15*x 
^4+84*x^3+208*x^2+192*x)*log(x^2+3*x)^2-5*x^3-35*x^2-60*x),x)
 

Output:

log( - sqrt(5) + log(x**2 + 3*x)*x + 4*log(x**2 + 3*x)) + log(sqrt(5) + lo 
g(x**2 + 3*x)*x + 4*log(x**2 + 3*x)) - 2*log(x + 4)