Integrand size = 88, antiderivative size = 23 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=2+\log \left (\frac {5}{(4+x)^2}-\log ^2\left (3 x+x^2\right )\right ) \]
Time = 0.73 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.91 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=-2 \log (4+x)+\log \left (5-16 \log ^2(x (3+x))-8 x \log ^2(x (3+x))-x^2 \log ^2(x (3+x))\right ) \]
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]
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 |
\(\displaystyle \int \frac {-10 x (x+3)-2 (2 x+3) (x+4)^3 \log (x (x+3))}{x (x+3) (x+4) \left (5-(x+4)^2 \log ^2(x (x+3))\right )}dx\) |
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]
3.15.29.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.18 (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 | \(-2 \ln \left (4+x \right )+\ln \left (\ln \left (x^{2}+3 x \right )^{2} x^{2}+8 \ln \left (x^{2}+3 x \right )^{2} x +16 \ln \left (x^{2}+3 x \right )^{2}-5\right )\) | \(50\) |
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)
Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.43 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=\log \left (\frac {{\left (x^{2} + 8 \, x + 16\right )} \log \left (x^{2} + 3 \, x\right )^{2} - 5}{x^{2} + 8 \, x + 16}\right ) \]
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=\
Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=\log {\left (\log {\left (x^{2} + 3 x \right )}^{2} - \frac {5}{x^{2} + 8 x + 16} \right )} \]
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)
Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (23) = 46\).
Time = 0.26 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.52 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=\log \left (\frac {{\left (x^{2} + 8 \, x + 16\right )} \log \left (x + 3\right )^{2} + 2 \, {\left (x^{2} + 8 \, x + 16\right )} \log \left (x + 3\right ) \log \left (x\right ) + {\left (x^{2} + 8 \, x + 16\right )} \log \left (x\right )^{2} - 5}{x^{2} + 8 \, x + 16}\right ) \]
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=\
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))
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).
Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.13 \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=\log \left (x^{2} \log \left (x^{2} + 3 \, x\right )^{2} + 8 \, x \log \left (x^{2} + 3 \, x\right )^{2} + 16 \, \log \left (x^{2} + 3 \, x\right )^{2} - 5\right ) - 2 \, \log \left (x + 4\right ) \]
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=\
Timed out. \[ \int \frac {30 x+10 x^2+\left (384+544 x+264 x^2+54 x^3+4 x^4\right ) \log \left (3 x+x^2\right )}{-60 x-35 x^2-5 x^3+\left (192 x+208 x^2+84 x^3+15 x^4+x^5\right ) \log ^2\left (3 x+x^2\right )} \, dx=-\int \frac {30\,x+\ln \left (x^2+3\,x\right )\,\left (4\,x^4+54\,x^3+264\,x^2+544\,x+384\right )+10\,x^2}{60\,x+35\,x^2+5\,x^3-{\ln \left (x^2+3\,x\right )}^2\,\left (x^5+15\,x^4+84\,x^3+208\,x^2+192\,x\right )} \,d x \]
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)