Integrand size = 32, antiderivative size = 579 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=-\frac {4 a b i (f h-e i)^3 x}{d f^4}+\frac {8 b^2 i (f h-e i)^3 x}{d f^4}+\frac {3 b^2 i^2 (f h-e i)^2 (e+f x)^2}{2 d f^5}+\frac {8 b^2 i^3 (f h-e i) (e+f x)^3}{27 d f^5}+\frac {b^2 i^4 (e+f x)^4}{32 d f^5}+\frac {7 b^2 (f h-e i)^4 \log ^2(e+f x)}{12 d f^5}-\frac {4 b^2 i (f h-e i)^3 (e+f x) \log (c (e+f x))}{d f^5}-\frac {4 b i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))}{d f^5}-\frac {3 b i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))}{d f^5}-\frac {8 b i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))}{9 d f^5}-\frac {b i^4 (e+f x)^4 (a+b \log (c (e+f x)))}{8 d f^5}-\frac {7 b (f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))}{6 d f^5}+\frac {2 i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))^2}{d f^5}+\frac {i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2}{2 d f^5}+\frac {(f h-e i) (h+i x)^3 (a+b \log (c (e+f x)))^2}{3 d f^2}+\frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{4 d f}+\frac {(f h-e i)^4 (a+b \log (c (e+f x)))^3}{3 b d f^5} \]
-4*a*b*i*(-e*i+f*h)^3*x/d/f^4+8*b^2*i*(-e*i+f*h)^3*x/d/f^4+3/2*b^2*i^2*(-e *i+f*h)^2*(f*x+e)^2/d/f^5+8/27*b^2*i^3*(-e*i+f*h)*(f*x+e)^3/d/f^5+1/32*b^2 *i^4*(f*x+e)^4/d/f^5+7/12*b^2*(-e*i+f*h)^4*ln(f*x+e)^2/d/f^5-4*b^2*i*(-e*i +f*h)^3*(f*x+e)*ln(c*(f*x+e))/d/f^5-4*b*i*(-e*i+f*h)^3*(f*x+e)*(a+b*ln(c*( f*x+e)))/d/f^5-3*b*i^2*(-e*i+f*h)^2*(f*x+e)^2*(a+b*ln(c*(f*x+e)))/d/f^5-8/ 9*b*i^3*(-e*i+f*h)*(f*x+e)^3*(a+b*ln(c*(f*x+e)))/d/f^5-1/8*b*i^4*(f*x+e)^4 *(a+b*ln(c*(f*x+e)))/d/f^5-7/6*b*(-e*i+f*h)^4*ln(f*x+e)*(a+b*ln(c*(f*x+e)) )/d/f^5+2*i*(-e*i+f*h)^3*(f*x+e)*(a+b*ln(c*(f*x+e)))^2/d/f^5+1/2*i^2*(-e*i +f*h)^2*(f*x+e)^2*(a+b*ln(c*(f*x+e)))^2/d/f^5+1/3*(-e*i+f*h)*(i*x+h)^3*(a+ b*ln(c*(f*x+e)))^2/d/f^2+1/4*(i*x+h)^4*(a+b*ln(c*(f*x+e)))^2/d/f+1/3*(-e*i +f*h)^4*(a+b*ln(c*(f*x+e)))^3/b/d/f^5
Time = 0.37 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.65 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {3456 i (f h-e i)^3 (e+f x) (a+b \log (c (e+f x)))^2+2592 i^2 (f h-e i)^2 (e+f x)^2 (a+b \log (c (e+f x)))^2+1152 i^3 (f h-e i) (e+f x)^3 (a+b \log (c (e+f x)))^2+216 i^4 (e+f x)^4 (a+b \log (c (e+f x)))^2+\frac {288 (f h-e i)^4 (a+b \log (c (e+f x)))^3}{b}-6912 b i (f h-e i)^3 ((a-b) f x+b (e+f x) \log (c (e+f x)))+1296 b i^2 (f h-e i)^2 \left (b f x (2 e+f x)-2 (e+f x)^2 (a+b \log (c (e+f x)))\right )+256 b i^3 (f h-e i) \left (b f x \left (3 e^2+3 e f x+f^2 x^2\right )-3 (e+f x)^3 (a+b \log (c (e+f x)))\right )+27 b i^4 \left (b f x \left (4 e^3+6 e^2 f x+4 e f^2 x^2+f^3 x^3\right )-4 (e+f x)^4 (a+b \log (c (e+f x)))\right )}{864 d f^5} \]
(3456*i*(f*h - e*i)^3*(e + f*x)*(a + b*Log[c*(e + f*x)])^2 + 2592*i^2*(f*h - e*i)^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2 + 1152*i^3*(f*h - e*i)*(e + f*x)^3*(a + b*Log[c*(e + f*x)])^2 + 216*i^4*(e + f*x)^4*(a + b*Log[c*(e + f*x)])^2 + (288*(f*h - e*i)^4*(a + b*Log[c*(e + f*x)])^3)/b - 6912*b*i* (f*h - e*i)^3*((a - b)*f*x + b*(e + f*x)*Log[c*(e + f*x)]) + 1296*b*i^2*(f *h - e*i)^2*(b*f*x*(2*e + f*x) - 2*(e + f*x)^2*(a + b*Log[c*(e + f*x)])) + 256*b*i^3*(f*h - e*i)*(b*f*x*(3*e^2 + 3*e*f*x + f^2*x^2) - 3*(e + f*x)^3* (a + b*Log[c*(e + f*x)])) + 27*b*i^4*(b*f*x*(4*e^3 + 6*e^2*f*x + 4*e*f^2*x ^2 + f^3*x^3) - 4*(e + f*x)^4*(a + b*Log[c*(e + f*x)])))/(864*d*f^5)
Time = 2.84 (sec) , antiderivative size = 838, normalized size of antiderivative = 1.45, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {2858, 27, 2788, 2756, 2772, 2009, 2788, 2756, 2772, 2009, 2788, 2767, 2009, 2788, 2733, 2009, 2739, 15}
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.
| \(\displaystyle \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {\int \frac {\left (f \left (h-\frac {e i}{f}\right )+i (e+f x)\right )^4 (a+b \log (c (e+f x)))^2}{d f^4 (e+f x)}d(e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(f h-e i+i (e+f x))^4 (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)}{d f^5}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {i \int (f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))^2d(e+f x)+(f h-e i) \int \frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)}{d f^5}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {i \left (\frac {(i (e+f x)-e i+f h)^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \int \frac {(f h-e i+i (e+f x))^4 (a+b \log (c (e+f x)))}{e+f x}d(e+f x)}{2 i}\right )+(f h-e i) \int \frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)}{d f^5}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {i \left (\frac {(i (e+f x)-e i+f h)^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (-b \int \left (\frac {1}{4} (e+f x)^3 i^4+\frac {4}{3} (f h-e i) (e+f x)^2 i^3+3 (f h-e i)^2 (e+f x) i^2+4 (f h-e i)^3 i+\frac {(f h-e i)^4 \log (e+f x)}{e+f x}\right )d(e+f x)+\frac {4}{3} i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))+3 i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))+(f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))+4 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))\right )}{2 i}\right )+(f h-e i) \int \frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)}{d f^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(f h-e i) \int \frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)+i \left (\frac {(i (e+f x)-e i+f h)^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\frac {4}{3} i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))+3 i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))+(f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))+4 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))-b \left (\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+4 i (e+f x) (f h-e i)^3+\frac {1}{2} (f h-e i)^4 \log ^2(e+f x)+\frac {1}{16} i^4 (e+f x)^4\right )\right )}{2 i}\right )}{d f^5}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {(f h-e i) \left (i \int (f h-e i+i (e+f x))^2 (a+b \log (c (e+f x)))^2d(e+f x)+(f h-e i) \int \frac {(f h-e i+i (e+f x))^2 (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)\right )+i \left (\frac {(i (e+f x)-e i+f h)^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\frac {4}{3} i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))+3 i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))+(f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))+4 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))-b \left (\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+4 i (e+f x) (f h-e i)^3+\frac {1}{2} (f h-e i)^4 \log ^2(e+f x)+\frac {1}{16} i^4 (e+f x)^4\right )\right )}{2 i}\right )}{d f^5}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {(f h-e i) \left (i \left (\frac {(i (e+f x)-e i+f h)^3 (a+b \log (c (e+f x)))^2}{3 i}-\frac {2 b \int \frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))}{e+f x}d(e+f x)}{3 i}\right )+(f h-e i) \int \frac {(f h-e i+i (e+f x))^2 (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)\right )+i \left (\frac {(i (e+f x)-e i+f h)^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\frac {4}{3} i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))+3 i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))+(f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))+4 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))-b \left (\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+4 i (e+f x) (f h-e i)^3+\frac {1}{2} (f h-e i)^4 \log ^2(e+f x)+\frac {1}{16} i^4 (e+f x)^4\right )\right )}{2 i}\right )}{d f^5}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {(f h-e i) \left (i \left (\frac {(i (e+f x)-e i+f h)^3 (a+b \log (c (e+f x)))^2}{3 i}-\frac {2 b \left (-b \int \left (\frac {1}{3} (e+f x)^2 i^3+\frac {3}{2} (f h-e i) (e+f x) i^2+3 (f h-e i)^2 i+\frac {(f h-e i)^3 \log (e+f x)}{e+f x}\right )d(e+f x)+\frac {3}{2} i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))+(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))+3 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))\right )}{3 i}\right )+(f h-e i) \int \frac {(f h-e i+i (e+f x))^2 (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)\right )+i \left (\frac {(i (e+f x)-e i+f h)^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\frac {4}{3} i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))+3 i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))+(f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))+4 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))-b \left (\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+4 i (e+f x) (f h-e i)^3+\frac {1}{2} (f h-e i)^4 \log ^2(e+f x)+\frac {1}{16} i^4 (e+f x)^4\right )\right )}{2 i}\right )}{d f^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(f h-e i) \left ((f h-e i) \int \frac {(f h-e i+i (e+f x))^2 (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)+i \left (\frac {(i (e+f x)-e i+f h)^3 (a+b \log (c (e+f x)))^2}{3 i}-\frac {2 b \left (\frac {3}{2} i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))+(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))+3 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))-b \left (\frac {3}{4} i^2 (e+f x)^2 (f h-e i)+3 i (e+f x) (f h-e i)^2+\frac {1}{2} (f h-e i)^3 \log ^2(e+f x)+\frac {1}{9} i^3 (e+f x)^3\right )\right )}{3 i}\right )\right )+i \left (\frac {(i (e+f x)-e i+f h)^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\frac {4}{3} i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))+3 i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))+(f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))+4 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))-b \left (\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+4 i (e+f x) (f h-e i)^3+\frac {1}{2} (f h-e i)^4 \log ^2(e+f x)+\frac {1}{16} i^4 (e+f x)^4\right )\right )}{2 i}\right )}{d f^5}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {(f h-e i) \left ((f h-e i) \left (i \int (f h-e i+i (e+f x)) (a+b \log (c (e+f x)))^2d(e+f x)+(f h-e i) \int \frac {(f h-e i+i (e+f x)) (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)\right )+i \left (\frac {(i (e+f x)-e i+f h)^3 (a+b \log (c (e+f x)))^2}{3 i}-\frac {2 b \left (\frac {3}{2} i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))+(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))+3 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))-b \left (\frac {3}{4} i^2 (e+f x)^2 (f h-e i)+3 i (e+f x) (f h-e i)^2+\frac {1}{2} (f h-e i)^3 \log ^2(e+f x)+\frac {1}{9} i^3 (e+f x)^3\right )\right )}{3 i}\right )\right )+i \left (\frac {(i (e+f x)-e i+f h)^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\frac {4}{3} i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))+3 i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))+(f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))+4 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))-b \left (\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+4 i (e+f x) (f h-e i)^3+\frac {1}{2} (f h-e i)^4 \log ^2(e+f x)+\frac {1}{16} i^4 (e+f x)^4\right )\right )}{2 i}\right )}{d f^5}\) |
| \(\Big \downarrow \) 2767 |
| \(\displaystyle \frac {(f h-e i) \left ((f h-e i) \left ((f h-e i) \int \frac {(f h-e i+i (e+f x)) (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)+i \int \left (f h \left (1-\frac {e i}{f h}\right ) (a+b \log (c (e+f x)))^2+i (e+f x) (a+b \log (c (e+f x)))^2\right )d(e+f x)\right )+i \left (\frac {(i (e+f x)-e i+f h)^3 (a+b \log (c (e+f x)))^2}{3 i}-\frac {2 b \left (\frac {3}{2} i^2 (e+f x)^2 (f h-e i) (a+b \log (c (e+f x)))+(f h-e i)^3 \log (e+f x) (a+b \log (c (e+f x)))+3 i (e+f x) (f h-e i)^2 (a+b \log (c (e+f x)))+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))-b \left (\frac {3}{4} i^2 (e+f x)^2 (f h-e i)+3 i (e+f x) (f h-e i)^2+\frac {1}{2} (f h-e i)^3 \log ^2(e+f x)+\frac {1}{9} i^3 (e+f x)^3\right )\right )}{3 i}\right )\right )+i \left (\frac {(i (e+f x)-e i+f h)^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\frac {4}{3} i^3 (e+f x)^3 (f h-e i) (a+b \log (c (e+f x)))+3 i^2 (e+f x)^2 (f h-e i)^2 (a+b \log (c (e+f x)))+(f h-e i)^4 \log (e+f x) (a+b \log (c (e+f x)))+4 i (e+f x) (f h-e i)^3 (a+b \log (c (e+f x)))+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))-b \left (\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+4 i (e+f x) (f h-e i)^3+\frac {1}{2} (f h-e i)^4 \log ^2(e+f x)+\frac {1}{16} i^4 (e+f x)^4\right )\right )}{2 i}\right )}{d f^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i \left (\frac {(f h-e i+i (e+f x))^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^4+4 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)^2+\frac {4}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^4+4 i (e+f x) (f h-e i)^3+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {1}{16} i^4 (e+f x)^4\right )+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))\right )}{2 i}\right )+(f h-e i) \left (i \left (\frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))^2}{3 i}-\frac {2 b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^2+\frac {3}{2} i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^3+3 i (e+f x) (f h-e i)^2+\frac {3}{4} i^2 (e+f x)^2 (f h-e i)+\frac {1}{9} i^3 (e+f x)^3\right )+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))\right )}{3 i}\right )+(f h-e i) \left (i \left (\frac {1}{4} i (e+f x)^2 b^2+2 (f h-e i) (e+f x) b^2-2 (f h-e i) (e+f x) \log (c (e+f x)) b^2-2 a (f h-e i) (e+f x) b-\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x))) b+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2+(f h-e i) (e+f x) (a+b \log (c (e+f x)))^2\right )+(f h-e i) \int \frac {(f h-e i+i (e+f x)) (a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)\right )\right )}{d f^5}\) |
| \(\Big \downarrow \) 2788 |
| \(\displaystyle \frac {i \left (\frac {(f h-e i+i (e+f x))^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^4+4 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)^2+\frac {4}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^4+4 i (e+f x) (f h-e i)^3+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {1}{16} i^4 (e+f x)^4\right )+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))\right )}{2 i}\right )+(f h-e i) \left (i \left (\frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))^2}{3 i}-\frac {2 b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^2+\frac {3}{2} i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^3+3 i (e+f x) (f h-e i)^2+\frac {3}{4} i^2 (e+f x)^2 (f h-e i)+\frac {1}{9} i^3 (e+f x)^3\right )+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))\right )}{3 i}\right )+(f h-e i) \left (i \left (\frac {1}{4} i (e+f x)^2 b^2+2 (f h-e i) (e+f x) b^2-2 (f h-e i) (e+f x) \log (c (e+f x)) b^2-2 a (f h-e i) (e+f x) b-\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x))) b+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2+(f h-e i) (e+f x) (a+b \log (c (e+f x)))^2\right )+(f h-e i) \left (i \int (a+b \log (c (e+f x)))^2d(e+f x)+(f h-e i) \int \frac {(a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)\right )\right )\right )}{d f^5}\) |
| \(\Big \downarrow \) 2733 |
| \(\displaystyle \frac {i \left (\frac {(f h-e i+i (e+f x))^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^4+4 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)^2+\frac {4}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^4+4 i (e+f x) (f h-e i)^3+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {1}{16} i^4 (e+f x)^4\right )+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))\right )}{2 i}\right )+(f h-e i) \left (i \left (\frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))^2}{3 i}-\frac {2 b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^2+\frac {3}{2} i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^3+3 i (e+f x) (f h-e i)^2+\frac {3}{4} i^2 (e+f x)^2 (f h-e i)+\frac {1}{9} i^3 (e+f x)^3\right )+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))\right )}{3 i}\right )+(f h-e i) \left (i \left (\frac {1}{4} i (e+f x)^2 b^2+2 (f h-e i) (e+f x) b^2-2 (f h-e i) (e+f x) \log (c (e+f x)) b^2-2 a (f h-e i) (e+f x) b-\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x))) b+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2+(f h-e i) (e+f x) (a+b \log (c (e+f x)))^2\right )+(f h-e i) \left (i \left ((e+f x) (a+b \log (c (e+f x)))^2-2 b \int (a+b \log (c (e+f x)))d(e+f x)\right )+(f h-e i) \int \frac {(a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)\right )\right )\right )}{d f^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i \left (\frac {(f h-e i+i (e+f x))^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^4+4 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)^2+\frac {4}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^4+4 i (e+f x) (f h-e i)^3+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {1}{16} i^4 (e+f x)^4\right )+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))\right )}{2 i}\right )+(f h-e i) \left (i \left (\frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))^2}{3 i}-\frac {2 b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^2+\frac {3}{2} i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^3+3 i (e+f x) (f h-e i)^2+\frac {3}{4} i^2 (e+f x)^2 (f h-e i)+\frac {1}{9} i^3 (e+f x)^3\right )+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))\right )}{3 i}\right )+(f h-e i) \left (i \left (\frac {1}{4} i (e+f x)^2 b^2+2 (f h-e i) (e+f x) b^2-2 (f h-e i) (e+f x) \log (c (e+f x)) b^2-2 a (f h-e i) (e+f x) b-\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x))) b+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2+(f h-e i) (e+f x) (a+b \log (c (e+f x)))^2\right )+(f h-e i) \left (i \left ((e+f x) (a+b \log (c (e+f x)))^2-2 b (a (e+f x)-b (e+f x)+b \log (c (e+f x)) (e+f x))\right )+(f h-e i) \int \frac {(a+b \log (c (e+f x)))^2}{e+f x}d(e+f x)\right )\right )\right )}{d f^5}\) |
| \(\Big \downarrow \) 2739 |
| \(\displaystyle \frac {i \left (\frac {(f h-e i+i (e+f x))^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^4+4 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)^2+\frac {4}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^4+4 i (e+f x) (f h-e i)^3+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {1}{16} i^4 (e+f x)^4\right )+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))\right )}{2 i}\right )+(f h-e i) \left (i \left (\frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))^2}{3 i}-\frac {2 b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^2+\frac {3}{2} i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^3+3 i (e+f x) (f h-e i)^2+\frac {3}{4} i^2 (e+f x)^2 (f h-e i)+\frac {1}{9} i^3 (e+f x)^3\right )+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))\right )}{3 i}\right )+(f h-e i) \left (i \left (\frac {1}{4} i (e+f x)^2 b^2+2 (f h-e i) (e+f x) b^2-2 (f h-e i) (e+f x) \log (c (e+f x)) b^2-2 a (f h-e i) (e+f x) b-\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x))) b+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2+(f h-e i) (e+f x) (a+b \log (c (e+f x)))^2\right )+(f h-e i) \left (i \left ((e+f x) (a+b \log (c (e+f x)))^2-2 b (a (e+f x)-b (e+f x)+b \log (c (e+f x)) (e+f x))\right )+\frac {(f h-e i) \int (a+b \log (c (e+f x)))^2d(a+b \log (c (e+f x)))}{b}\right )\right )\right )}{d f^5}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {i \left (\frac {(f h-e i+i (e+f x))^4 (a+b \log (c (e+f x)))^2}{4 i}-\frac {b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^4+4 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)^2+\frac {4}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^4+4 i (e+f x) (f h-e i)^3+\frac {3}{2} i^2 (e+f x)^2 (f h-e i)^2+\frac {4}{9} i^3 (e+f x)^3 (f h-e i)+\frac {1}{16} i^4 (e+f x)^4\right )+\frac {1}{4} i^4 (e+f x)^4 (a+b \log (c (e+f x)))\right )}{2 i}\right )+(f h-e i) \left (i \left (\frac {(f h-e i+i (e+f x))^3 (a+b \log (c (e+f x)))^2}{3 i}-\frac {2 b \left (\log (e+f x) (a+b \log (c (e+f x))) (f h-e i)^3+3 i (e+f x) (a+b \log (c (e+f x))) (f h-e i)^2+\frac {3}{2} i^2 (e+f x)^2 (a+b \log (c (e+f x))) (f h-e i)-b \left (\frac {1}{2} \log ^2(e+f x) (f h-e i)^3+3 i (e+f x) (f h-e i)^2+\frac {3}{4} i^2 (e+f x)^2 (f h-e i)+\frac {1}{9} i^3 (e+f x)^3\right )+\frac {1}{3} i^3 (e+f x)^3 (a+b \log (c (e+f x)))\right )}{3 i}\right )+(f h-e i) \left (i \left (\frac {1}{4} i (e+f x)^2 b^2+2 (f h-e i) (e+f x) b^2-2 (f h-e i) (e+f x) \log (c (e+f x)) b^2-2 a (f h-e i) (e+f x) b-\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x))) b+\frac {1}{2} i (e+f x)^2 (a+b \log (c (e+f x)))^2+(f h-e i) (e+f x) (a+b \log (c (e+f x)))^2\right )+(f h-e i) \left (\frac {(f h-e i) (a+b \log (c (e+f x)))^3}{3 b}+i \left ((e+f x) (a+b \log (c (e+f x)))^2-2 b (a (e+f x)-b (e+f x)+b \log (c (e+f x)) (e+f x))\right )\right )\right )\right )}{d f^5}\) |
(i*(((f*h - e*i + i*(e + f*x))^4*(a + b*Log[c*(e + f*x)])^2)/(4*i) - (b*(- (b*(4*i*(f*h - e*i)^3*(e + f*x) + (3*i^2*(f*h - e*i)^2*(e + f*x)^2)/2 + (4 *i^3*(f*h - e*i)*(e + f*x)^3)/9 + (i^4*(e + f*x)^4)/16 + ((f*h - e*i)^4*Lo g[e + f*x]^2)/2)) + 4*i*(f*h - e*i)^3*(e + f*x)*(a + b*Log[c*(e + f*x)]) + 3*i^2*(f*h - e*i)^2*(e + f*x)^2*(a + b*Log[c*(e + f*x)]) + (4*i^3*(f*h - e*i)*(e + f*x)^3*(a + b*Log[c*(e + f*x)]))/3 + (i^4*(e + f*x)^4*(a + b*Log [c*(e + f*x)]))/4 + (f*h - e*i)^4*Log[e + f*x]*(a + b*Log[c*(e + f*x)])))/ (2*i)) + (f*h - e*i)*(i*(((f*h - e*i + i*(e + f*x))^3*(a + b*Log[c*(e + f* x)])^2)/(3*i) - (2*b*(-(b*(3*i*(f*h - e*i)^2*(e + f*x) + (3*i^2*(f*h - e*i )*(e + f*x)^2)/4 + (i^3*(e + f*x)^3)/9 + ((f*h - e*i)^3*Log[e + f*x]^2)/2) ) + 3*i*(f*h - e*i)^2*(e + f*x)*(a + b*Log[c*(e + f*x)]) + (3*i^2*(f*h - e *i)*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/2 + (i^3*(e + f*x)^3*(a + b*Log[ c*(e + f*x)]))/3 + (f*h - e*i)^3*Log[e + f*x]*(a + b*Log[c*(e + f*x)])))/( 3*i)) + (f*h - e*i)*(i*(-2*a*b*(f*h - e*i)*(e + f*x) + 2*b^2*(f*h - e*i)*( e + f*x) + (b^2*i*(e + f*x)^2)/4 - 2*b^2*(f*h - e*i)*(e + f*x)*Log[c*(e + f*x)] - (b*i*(e + f*x)^2*(a + b*Log[c*(e + f*x)]))/2 + (f*h - e*i)*(e + f* x)*(a + b*Log[c*(e + f*x)])^2 + (i*(e + f*x)^2*(a + b*Log[c*(e + f*x)])^2) /2) + (f*h - e*i)*(((f*h - e*i)*(a + b*Log[c*(e + f*x)])^3)/(3*b) + i*((e + f*x)*(a + b*Log[c*(e + f*x)])^2 - 2*b*(a*(e + f*x) - b*(e + f*x) + b*(e + f*x)*Log[c*(e + f*x)]))))))/(d*f^5)
3.2.83.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b *Log[c*x^n])^p, x] - Simp[b*n*p Int[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Simp[1/( b*n) Subst[Int[x^p, x], x, a + b*Log[c*x^n]], x] /; FreeQ[{a, b, c, n, p} , x]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^( q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (d + e*x ^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[r]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.)) /(x_), x_Symbol] :> Simp[d Int[(d + e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x) , x], x] + Simp[e Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /; F reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Leaf count of result is larger than twice the leaf count of optimal. \(1134\) vs. \(2(557)=1114\).
Time = 0.92 (sec) , antiderivative size = 1135, normalized size of antiderivative = 1.96
| method | result | size |
| norman | \(\text {Expression too large to display}\) | \(1135\) |
| risch | \(\text {Expression too large to display}\) | \(1475\) |
| parts | \(\text {Expression too large to display}\) | \(1541\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1891\) |
| default | \(\text {Expression too large to display}\) | \(1891\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1912\) |
1/72*(72*a^2*e^4*i^4-288*a^2*e^3*f*h*i^3+432*a^2*e^2*f^2*h^2*i^2-288*a^2*e *f^3*h^3*i+72*a^2*f^4*h^4-300*a*b*e^4*i^4+1056*a*b*e^3*f*h*i^3-1296*a*b*e^ 2*f^2*h^2*i^2+576*a*b*e*f^3*h^3*i+415*b^2*e^4*i^4-1360*b^2*e^3*f*h*i^3+151 2*b^2*e^2*f^2*h^2*i^2-576*b^2*e*f^3*h^3*i)/d/f^5*ln(c*(f*x+e))+1/12*b*(12* a*e^4*i^4-48*a*e^3*f*h*i^3+72*a*e^2*f^2*h^2*i^2-48*a*e*f^3*h^3*i+12*a*f^4* h^4-25*b*e^4*i^4+88*b*e^3*f*h*i^3-108*b*e^2*f^2*h^2*i^2+48*b*e*f^3*h^3*i)/ d/f^5*ln(c*(f*x+e))^2+1/3*b^2*(e^4*i^4-4*e^3*f*h*i^3+6*e^2*f^2*h^2*i^2-4*e *f^3*h^3*i+f^4*h^4)/d/f^5*ln(c*(f*x+e))^3-1/72*i*(72*a^2*e^3*i^3-288*a^2*e ^2*f*h*i^2+432*a^2*e*f^2*h^2*i-288*a^2*f^3*h^3-300*a*b*e^3*i^3+1056*a*b*e^ 2*f*h*i^2-1296*a*b*e*f^2*h^2*i+576*a*b*f^3*h^3+415*b^2*e^3*i^3-1360*b^2*e^ 2*f*h*i^2+1512*b^2*e*f^2*h^2*i-576*b^2*f^3*h^3)/d/f^4*x+1/144*i^2*(72*a^2* e^2*i^2-288*a^2*e*f*h*i+432*a^2*f^2*h^2-156*a*b*e^2*i^2+480*a*b*e*f*h*i-43 2*a*b*f^2*h^2+115*b^2*e^2*i^2-304*b^2*e*f*h*i+216*b^2*f^2*h^2)/d/f^3*x^2-1 /216*i^3*(72*a^2*e*i-288*a^2*f*h-84*a*b*e*i+192*a*b*f*h+37*b^2*e*i-64*b^2* f*h)/f^2/d*x^3+1/32*i^4*(8*a^2-4*a*b+b^2)/d/f*x^4+1/4*b^2*i^4/d/f*x^4*ln(c *(f*x+e))^2-1/6*b*i*(12*a*e^3*i^3-48*a*e^2*f*h*i^2+72*a*e*f^2*h^2*i-48*a*f ^3*h^3-25*b*e^3*i^3+88*b*e^2*f*h*i^2-108*b*e*f^2*h^2*i+48*b*f^3*h^3)/d/f^4 *x*ln(c*(f*x+e))+1/12*b*i^2*(12*a*e^2*i^2-48*a*e*f*h*i+72*a*f^2*h^2-13*b*e ^2*i^2+40*b*e*f*h*i-36*b*f^2*h^2)/d/f^3*x^2*ln(c*(f*x+e))-1/18*b*i^3*(12*a *e*i-48*a*f*h-7*b*e*i+16*b*f*h)/d/f^2*x^3*ln(c*(f*x+e))+1/8*b*i^4*(4*a-...
Time = 0.30 (sec) , antiderivative size = 939, normalized size of antiderivative = 1.62 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\frac {27 \, {\left (8 \, a^{2} - 4 \, a b + b^{2}\right )} f^{4} i^{4} x^{4} + 4 \, {\left (32 \, {\left (9 \, a^{2} - 6 \, a b + 2 \, b^{2}\right )} f^{4} h i^{3} - {\left (72 \, a^{2} - 84 \, a b + 37 \, b^{2}\right )} e f^{3} i^{4}\right )} x^{3} + 288 \, {\left (b^{2} f^{4} h^{4} - 4 \, b^{2} e f^{3} h^{3} i + 6 \, b^{2} e^{2} f^{2} h^{2} i^{2} - 4 \, b^{2} e^{3} f h i^{3} + b^{2} e^{4} i^{4}\right )} \log \left (c f x + c e\right )^{3} + 6 \, {\left (216 \, {\left (2 \, a^{2} - 2 \, a b + b^{2}\right )} f^{4} h^{2} i^{2} - 16 \, {\left (18 \, a^{2} - 30 \, a b + 19 \, b^{2}\right )} e f^{3} h i^{3} + {\left (72 \, a^{2} - 156 \, a b + 115 \, b^{2}\right )} e^{2} f^{2} i^{4}\right )} x^{2} + 72 \, {\left (3 \, b^{2} f^{4} i^{4} x^{4} + 12 \, a b f^{4} h^{4} - 48 \, {\left (a b - b^{2}\right )} e f^{3} h^{3} i + 36 \, {\left (2 \, a b - 3 \, b^{2}\right )} e^{2} f^{2} h^{2} i^{2} - 8 \, {\left (6 \, a b - 11 \, b^{2}\right )} e^{3} f h i^{3} + {\left (12 \, a b - 25 \, b^{2}\right )} e^{4} i^{4} + 4 \, {\left (4 \, b^{2} f^{4} h i^{3} - b^{2} e f^{3} i^{4}\right )} x^{3} + 6 \, {\left (6 \, b^{2} f^{4} h^{2} i^{2} - 4 \, b^{2} e f^{3} h i^{3} + b^{2} e^{2} f^{2} i^{4}\right )} x^{2} + 12 \, {\left (4 \, b^{2} f^{4} h^{3} i - 6 \, b^{2} e f^{3} h^{2} i^{2} + 4 \, b^{2} e^{2} f^{2} h i^{3} - b^{2} e^{3} f i^{4}\right )} x\right )} \log \left (c f x + c e\right )^{2} + 12 \, {\left (288 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} f^{4} h^{3} i - 216 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e f^{3} h^{2} i^{2} + 16 \, {\left (18 \, a^{2} - 66 \, a b + 85 \, b^{2}\right )} e^{2} f^{2} h i^{3} - {\left (72 \, a^{2} - 300 \, a b + 415 \, b^{2}\right )} e^{3} f i^{4}\right )} x + 12 \, {\left (9 \, {\left (4 \, a b - b^{2}\right )} f^{4} i^{4} x^{4} + 72 \, a^{2} f^{4} h^{4} - 288 \, {\left (a^{2} - 2 \, a b + 2 \, b^{2}\right )} e f^{3} h^{3} i + 216 \, {\left (2 \, a^{2} - 6 \, a b + 7 \, b^{2}\right )} e^{2} f^{2} h^{2} i^{2} - 16 \, {\left (18 \, a^{2} - 66 \, a b + 85 \, b^{2}\right )} e^{3} f h i^{3} + {\left (72 \, a^{2} - 300 \, a b + 415 \, b^{2}\right )} e^{4} i^{4} + 4 \, {\left (16 \, {\left (3 \, a b - b^{2}\right )} f^{4} h i^{3} - {\left (12 \, a b - 7 \, b^{2}\right )} e f^{3} i^{4}\right )} x^{3} + 6 \, {\left (36 \, {\left (2 \, a b - b^{2}\right )} f^{4} h^{2} i^{2} - 8 \, {\left (6 \, a b - 5 \, b^{2}\right )} e f^{3} h i^{3} + {\left (12 \, a b - 13 \, b^{2}\right )} e^{2} f^{2} i^{4}\right )} x^{2} + 12 \, {\left (48 \, {\left (a b - b^{2}\right )} f^{4} h^{3} i - 36 \, {\left (2 \, a b - 3 \, b^{2}\right )} e f^{3} h^{2} i^{2} + 8 \, {\left (6 \, a b - 11 \, b^{2}\right )} e^{2} f^{2} h i^{3} - {\left (12 \, a b - 25 \, b^{2}\right )} e^{3} f i^{4}\right )} x\right )} \log \left (c f x + c e\right )}{864 \, d f^{5}} \]
1/864*(27*(8*a^2 - 4*a*b + b^2)*f^4*i^4*x^4 + 4*(32*(9*a^2 - 6*a*b + 2*b^2 )*f^4*h*i^3 - (72*a^2 - 84*a*b + 37*b^2)*e*f^3*i^4)*x^3 + 288*(b^2*f^4*h^4 - 4*b^2*e*f^3*h^3*i + 6*b^2*e^2*f^2*h^2*i^2 - 4*b^2*e^3*f*h*i^3 + b^2*e^4 *i^4)*log(c*f*x + c*e)^3 + 6*(216*(2*a^2 - 2*a*b + b^2)*f^4*h^2*i^2 - 16*( 18*a^2 - 30*a*b + 19*b^2)*e*f^3*h*i^3 + (72*a^2 - 156*a*b + 115*b^2)*e^2*f ^2*i^4)*x^2 + 72*(3*b^2*f^4*i^4*x^4 + 12*a*b*f^4*h^4 - 48*(a*b - b^2)*e*f^ 3*h^3*i + 36*(2*a*b - 3*b^2)*e^2*f^2*h^2*i^2 - 8*(6*a*b - 11*b^2)*e^3*f*h* i^3 + (12*a*b - 25*b^2)*e^4*i^4 + 4*(4*b^2*f^4*h*i^3 - b^2*e*f^3*i^4)*x^3 + 6*(6*b^2*f^4*h^2*i^2 - 4*b^2*e*f^3*h*i^3 + b^2*e^2*f^2*i^4)*x^2 + 12*(4* b^2*f^4*h^3*i - 6*b^2*e*f^3*h^2*i^2 + 4*b^2*e^2*f^2*h*i^3 - b^2*e^3*f*i^4) *x)*log(c*f*x + c*e)^2 + 12*(288*(a^2 - 2*a*b + 2*b^2)*f^4*h^3*i - 216*(2* a^2 - 6*a*b + 7*b^2)*e*f^3*h^2*i^2 + 16*(18*a^2 - 66*a*b + 85*b^2)*e^2*f^2 *h*i^3 - (72*a^2 - 300*a*b + 415*b^2)*e^3*f*i^4)*x + 12*(9*(4*a*b - b^2)*f ^4*i^4*x^4 + 72*a^2*f^4*h^4 - 288*(a^2 - 2*a*b + 2*b^2)*e*f^3*h^3*i + 216* (2*a^2 - 6*a*b + 7*b^2)*e^2*f^2*h^2*i^2 - 16*(18*a^2 - 66*a*b + 85*b^2)*e^ 3*f*h*i^3 + (72*a^2 - 300*a*b + 415*b^2)*e^4*i^4 + 4*(16*(3*a*b - b^2)*f^4 *h*i^3 - (12*a*b - 7*b^2)*e*f^3*i^4)*x^3 + 6*(36*(2*a*b - b^2)*f^4*h^2*i^2 - 8*(6*a*b - 5*b^2)*e*f^3*h*i^3 + (12*a*b - 13*b^2)*e^2*f^2*i^4)*x^2 + 12 *(48*(a*b - b^2)*f^4*h^3*i - 36*(2*a*b - 3*b^2)*e*f^3*h^2*i^2 + 8*(6*a*b - 11*b^2)*e^2*f^2*h*i^3 - (12*a*b - 25*b^2)*e^3*f*i^4)*x)*log(c*f*x + c*...
Leaf count of result is larger than twice the leaf count of optimal. 1479 vs. \(2 (534) = 1068\).
Time = 1.88 (sec) , antiderivative size = 1479, normalized size of antiderivative = 2.55 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\text {Too large to display} \]
x**4*(a**2*i**4/(4*d*f) - a*b*i**4/(8*d*f) + b**2*i**4/(32*d*f)) + x**3*(- a**2*e*i**4/(3*d*f**2) + 4*a**2*h*i**3/(3*d*f) + 7*a*b*e*i**4/(18*d*f**2) - 8*a*b*h*i**3/(9*d*f) - 37*b**2*e*i**4/(216*d*f**2) + 8*b**2*h*i**3/(27*d *f)) + x**2*(a**2*e**2*i**4/(2*d*f**3) - 2*a**2*e*h*i**3/(d*f**2) + 3*a**2 *h**2*i**2/(d*f) - 13*a*b*e**2*i**4/(12*d*f**3) + 10*a*b*e*h*i**3/(3*d*f** 2) - 3*a*b*h**2*i**2/(d*f) + 115*b**2*e**2*i**4/(144*d*f**3) - 19*b**2*e*h *i**3/(9*d*f**2) + 3*b**2*h**2*i**2/(2*d*f)) + x*(-a**2*e**3*i**4/(d*f**4) + 4*a**2*e**2*h*i**3/(d*f**3) - 6*a**2*e*h**2*i**2/(d*f**2) + 4*a**2*h**3 *i/(d*f) + 25*a*b*e**3*i**4/(6*d*f**4) - 44*a*b*e**2*h*i**3/(3*d*f**3) + 1 8*a*b*e*h**2*i**2/(d*f**2) - 8*a*b*h**3*i/(d*f) - 415*b**2*e**3*i**4/(72*d *f**4) + 170*b**2*e**2*h*i**3/(9*d*f**3) - 21*b**2*e*h**2*i**2/(d*f**2) + 8*b**2*h**3*i/(d*f)) + (-144*a*b*e**3*i**4*x + 576*a*b*e**2*f*h*i**3*x + 7 2*a*b*e**2*f*i**4*x**2 - 864*a*b*e*f**2*h**2*i**2*x - 288*a*b*e*f**2*h*i** 3*x**2 - 48*a*b*e*f**2*i**4*x**3 + 576*a*b*f**3*h**3*i*x + 432*a*b*f**3*h* *2*i**2*x**2 + 192*a*b*f**3*h*i**3*x**3 + 36*a*b*f**3*i**4*x**4 + 300*b**2 *e**3*i**4*x - 1056*b**2*e**2*f*h*i**3*x - 78*b**2*e**2*f*i**4*x**2 + 1296 *b**2*e*f**2*h**2*i**2*x + 240*b**2*e*f**2*h*i**3*x**2 + 28*b**2*e*f**2*i* *4*x**3 - 576*b**2*f**3*h**3*i*x - 216*b**2*f**3*h**2*i**2*x**2 - 64*b**2* f**3*h*i**3*x**3 - 9*b**2*f**3*i**4*x**4)*log(c*(e + f*x))/(72*d*f**4) + ( b**2*e**4*i**4 - 4*b**2*e**3*f*h*i**3 + 6*b**2*e**2*f**2*h**2*i**2 - 4*...
Leaf count of result is larger than twice the leaf count of optimal. 1427 vs. \(2 (557) = 1114\).
Time = 0.27 (sec) , antiderivative size = 1427, normalized size of antiderivative = 2.46 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\text {Too large to display} \]
8*a*b*h^3*i*(x/(d*f) - e*log(f*x + e)/(d*f^2))*log(c*f*x + c*e) + 1/6*a*b* i^4*(12*e^4*log(f*x + e)/(d*f^5) + (3*f^3*x^4 - 4*e*f^2*x^3 + 6*e^2*f*x^2 - 12*e^3*x)/(d*f^4))*log(c*f*x + c*e) - 4/3*a*b*h*i^3*(6*e^3*log(f*x + e)/ (d*f^4) - (2*f^2*x^3 - 3*e*f*x^2 + 6*e^2*x)/(d*f^3))*log(c*f*x + c*e) + 6* a*b*h^2*i^2*(2*e^2*log(f*x + e)/(d*f^3) + (f*x^2 - 2*e*x)/(d*f^2))*log(c*f *x + c*e) - a*b*h^4*(2*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) - (log(f*x + e)^2 + 2*log(f*x + e)*log(c))/(d*f)) + 4*a^2*h^3*i*(x/(d*f) - e*log(f*x + e)/(d*f^2)) + 1/12*a^2*i^4*(12*e^4*log(f*x + e)/(d*f^5) + (3*f^3*x^4 - 4 *e*f^2*x^3 + 6*e^2*f*x^2 - 12*e^3*x)/(d*f^4)) - 2/3*a^2*h*i^3*(6*e^3*log(f *x + e)/(d*f^4) - (2*f^2*x^3 - 3*e*f*x^2 + 6*e^2*x)/(d*f^3)) + 3*a^2*h^2*i ^2*(2*e^2*log(f*x + e)/(d*f^3) + (f*x^2 - 2*e*x)/(d*f^2)) + 1/3*b^2*h^4*lo g(c*f*x + c*e)^3/(d*f) + 2*a*b*h^4*log(c*f*x + c*e)*log(d*f*x + d*e)/(d*f) + a^2*h^4*log(d*f*x + d*e)/(d*f) + 4*(e*log(f*x + e)^2 - 2*f*x + 2*e*log( f*x + e))*a*b*h^3*i/(d*f^2) - 3*(f^2*x^2 + 2*e^2*log(f*x + e)^2 - 6*e*f*x + 6*e^2*log(f*x + e))*a*b*h^2*i^2/(d*f^3) - 4/3*(c^2*e*log(c*f*x + c*e)^3 - 3*(c*f*x + c*e)*(c*log(c*f*x + c*e)^2 - 2*c*log(c*f*x + c*e) + 2*c))*b^2 *h^3*i/(c^2*d*f^2) - 2/9*(4*f^3*x^3 - 15*e*f^2*x^2 - 18*e^3*log(f*x + e)^2 + 66*e^2*f*x - 66*e^3*log(f*x + e))*a*b*h*i^3/(d*f^4) - 1/72*(9*f^4*x^4 - 28*e*f^3*x^3 + 78*e^2*f^2*x^2 + 72*e^4*log(f*x + e)^2 - 300*e^3*f*x + 300 *e^4*log(f*x + e))*a*b*i^4/(d*f^5) + 1/2*(4*c^3*e^2*log(c*f*x + c*e)^3 ...
Leaf count of result is larger than twice the leaf count of optimal. 1213 vs. \(2 (557) = 1114\).
Time = 0.33 (sec) , antiderivative size = 1213, normalized size of antiderivative = 2.09 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx=\text {Too large to display} \]
1/32*(8*a^2*i^4 - 4*a*b*i^4 + b^2*i^4)*x^4/(d*f) + 1/12*(3*b^2*i^4*x^4/(d* f) + 4*(4*b^2*f*h*i^3 - b^2*e*i^4)*x^3/(d*f^2) + 6*(6*b^2*f^2*h^2*i^2 - 4* b^2*e*f*h*i^3 + b^2*e^2*i^4)*x^2/(d*f^3) + 12*(4*b^2*f^3*h^3*i - 6*b^2*e*f ^2*h^2*i^2 + 4*b^2*e^2*f*h*i^3 - b^2*e^3*i^4)*x/(d*f^4) + (12*a*b*f^4*h^4 - 48*a*b*e*f^3*h^3*i + 48*b^2*e*f^3*h^3*i + 72*a*b*e^2*f^2*h^2*i^2 - 108*b ^2*e^2*f^2*h^2*i^2 - 48*a*b*e^3*f*h*i^3 + 88*b^2*e^3*f*h*i^3 + 12*a*b*e^4* i^4 - 25*b^2*e^4*i^4)/(d*f^5))*log(c*f*x + c*e)^2 + 1/72*(9*(4*a*b*i^4 - b ^2*i^4)*x^4/(d*f) + 4*(48*a*b*f*h*i^3 - 16*b^2*f*h*i^3 - 12*a*b*e*i^4 + 7* b^2*e*i^4)*x^3/(d*f^2) + 6*(72*a*b*f^2*h^2*i^2 - 36*b^2*f^2*h^2*i^2 - 48*a *b*e*f*h*i^3 + 40*b^2*e*f*h*i^3 + 12*a*b*e^2*i^4 - 13*b^2*e^2*i^4)*x^2/(d* f^3) + 12*(48*a*b*f^3*h^3*i - 48*b^2*f^3*h^3*i - 72*a*b*e*f^2*h^2*i^2 + 10 8*b^2*e*f^2*h^2*i^2 + 48*a*b*e^2*f*h*i^3 - 88*b^2*e^2*f*h*i^3 - 12*a*b*e^3 *i^4 + 25*b^2*e^3*i^4)*x/(d*f^4))*log(c*f*x + c*e) + 1/216*(288*a^2*f*h*i^ 3 - 192*a*b*f*h*i^3 + 64*b^2*f*h*i^3 - 72*a^2*e*i^4 + 84*a*b*e*i^4 - 37*b^ 2*e*i^4)*x^3/(d*f^2) + 1/144*(432*a^2*f^2*h^2*i^2 - 432*a*b*f^2*h^2*i^2 + 216*b^2*f^2*h^2*i^2 - 288*a^2*e*f*h*i^3 + 480*a*b*e*f*h*i^3 - 304*b^2*e*f* h*i^3 + 72*a^2*e^2*i^4 - 156*a*b*e^2*i^4 + 115*b^2*e^2*i^4)*x^2/(d*f^3) + 1/3*(b^2*f^4*h^4 - 4*b^2*e*f^3*h^3*i + 6*b^2*e^2*f^2*h^2*i^2 - 4*b^2*e^3*f *h*i^3 + b^2*e^4*i^4)*log(c*f*x + c*e)^3/(d*f^5) + 1/72*(288*a^2*f^3*h^3*i - 576*a*b*f^3*h^3*i + 576*b^2*f^3*h^3*i - 432*a^2*e*f^2*h^2*i^2 + 1296...
Time = 2.07 (sec) , antiderivative size = 1346, normalized size of antiderivative = 2.32 \[ \int \frac {(h+i x)^4 (a+b \log (c (e+f x)))^2}{d e+d f x} \, dx={\ln \left (c\,\left (e+f\,x\right )\right )}^2\,\left (f\,\left (\frac {b^2\,i^4\,x^4}{4\,d\,f^2}-\frac {b^2\,i^3\,x^3\,\left (e\,i-4\,f\,h\right )}{3\,d\,f^3}-\frac {b^2\,i\,x\,\left (e^3\,i^3-4\,e^2\,f\,h\,i^2+6\,e\,f^2\,h^2\,i-4\,f^3\,h^3\right )}{d\,f^5}+\frac {b^2\,i^2\,x^2\,\left (e^2\,i^2-4\,e\,f\,h\,i+6\,f^2\,h^2\right )}{2\,d\,f^4}\right )+\frac {-25\,b^2\,e^4\,i^4+88\,b^2\,e^3\,f\,h\,i^3-108\,b^2\,e^2\,f^2\,h^2\,i^2+48\,b^2\,e\,f^3\,h^3\,i+12\,a\,b\,e^4\,i^4-48\,a\,b\,e^3\,f\,h\,i^3+72\,a\,b\,e^2\,f^2\,h^2\,i^2-48\,a\,b\,e\,f^3\,h^3\,i+12\,a\,b\,f^4\,h^4}{12\,d\,f^5}\right )-x^2\,\left (\frac {e\,\left (\frac {i^3\,\left (72\,a^2\,f\,h-7\,b^2\,e\,i+16\,b^2\,f\,h+12\,a\,b\,e\,i-48\,a\,b\,f\,h\right )}{18\,d\,f^2}-\frac {e\,i^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{8\,d\,f^2}\right )}{2\,f}-\frac {i^2\,\left (72\,a^2\,f^2\,h^2-12\,a\,b\,e^2\,i^2+48\,a\,b\,e\,f\,h\,i-72\,a\,b\,f^2\,h^2+13\,b^2\,e^2\,i^2-40\,b^2\,e\,f\,h\,i+36\,b^2\,f^2\,h^2\right )}{24\,d\,f^3}\right )+x^3\,\left (\frac {i^3\,\left (72\,a^2\,f\,h-7\,b^2\,e\,i+16\,b^2\,f\,h+12\,a\,b\,e\,i-48\,a\,b\,f\,h\right )}{54\,d\,f^2}-\frac {e\,i^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{24\,d\,f^2}\right )+x\,\left (\frac {288\,a^2\,f^3\,h^3\,i+144\,a\,b\,e^3\,i^4-576\,a\,b\,e^2\,f\,h\,i^3+864\,a\,b\,e\,f^2\,h^2\,i^2-576\,a\,b\,f^3\,h^3\,i-300\,b^2\,e^3\,i^4+1056\,b^2\,e^2\,f\,h\,i^3-1296\,b^2\,e\,f^2\,h^2\,i^2+576\,b^2\,f^3\,h^3\,i}{72\,d\,f^4}+\frac {e\,\left (\frac {e\,\left (\frac {i^3\,\left (72\,a^2\,f\,h-7\,b^2\,e\,i+16\,b^2\,f\,h+12\,a\,b\,e\,i-48\,a\,b\,f\,h\right )}{18\,d\,f^2}-\frac {e\,i^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{8\,d\,f^2}\right )}{f}-\frac {i^2\,\left (72\,a^2\,f^2\,h^2-12\,a\,b\,e^2\,i^2+48\,a\,b\,e\,f\,h\,i-72\,a\,b\,f^2\,h^2+13\,b^2\,e^2\,i^2-40\,b^2\,e\,f\,h\,i+36\,b^2\,f^2\,h^2\right )}{12\,d\,f^3}\right )}{f}\right )+f\,\ln \left (c\,\left (e+f\,x\right )\right )\,\left (\frac {x^3\,\left (7\,e\,b^2\,i^4-16\,f\,h\,b^2\,i^3-12\,a\,e\,b\,i^4+48\,a\,f\,h\,b\,i^3\right )}{18\,d\,f^3}-\frac {x^2\,\left (13\,b^2\,e^2\,i^4-40\,b^2\,e\,f\,h\,i^3+36\,b^2\,f^2\,h^2\,i^2-12\,a\,b\,e^2\,i^4+48\,a\,b\,e\,f\,h\,i^3-72\,a\,b\,f^2\,h^2\,i^2\right )}{12\,d\,f^4}+\frac {x\,\left (25\,b^2\,e^3\,i^4-88\,b^2\,e^2\,f\,h\,i^3+108\,b^2\,e\,f^2\,h^2\,i^2-48\,b^2\,f^3\,h^3\,i-12\,a\,b\,e^3\,i^4+48\,a\,b\,e^2\,f\,h\,i^3-72\,a\,b\,e\,f^2\,h^2\,i^2+48\,a\,b\,f^3\,h^3\,i\right )}{6\,d\,f^5}+\frac {b\,i^4\,x^4\,\left (4\,a-b\right )}{8\,d\,f^2}\right )+\frac {\ln \left (e+f\,x\right )\,\left (72\,a^2\,e^4\,i^4-288\,a^2\,e^3\,f\,h\,i^3+432\,a^2\,e^2\,f^2\,h^2\,i^2-288\,a^2\,e\,f^3\,h^3\,i+72\,a^2\,f^4\,h^4-300\,a\,b\,e^4\,i^4+1056\,a\,b\,e^3\,f\,h\,i^3-1296\,a\,b\,e^2\,f^2\,h^2\,i^2+576\,a\,b\,e\,f^3\,h^3\,i+415\,b^2\,e^4\,i^4-1360\,b^2\,e^3\,f\,h\,i^3+1512\,b^2\,e^2\,f^2\,h^2\,i^2-576\,b^2\,e\,f^3\,h^3\,i\right )}{72\,d\,f^5}+\frac {b^2\,{\ln \left (c\,\left (e+f\,x\right )\right )}^3\,\left (e^4\,i^4-4\,e^3\,f\,h\,i^3+6\,e^2\,f^2\,h^2\,i^2-4\,e\,f^3\,h^3\,i+f^4\,h^4\right )}{3\,d\,f^5}+\frac {i^4\,x^4\,\left (8\,a^2-4\,a\,b+b^2\right )}{32\,d\,f} \]
log(c*(e + f*x))^2*(f*((b^2*i^4*x^4)/(4*d*f^2) - (b^2*i^3*x^3*(e*i - 4*f*h ))/(3*d*f^3) - (b^2*i*x*(e^3*i^3 - 4*f^3*h^3 + 6*e*f^2*h^2*i - 4*e^2*f*h*i ^2))/(d*f^5) + (b^2*i^2*x^2*(e^2*i^2 + 6*f^2*h^2 - 4*e*f*h*i))/(2*d*f^4)) + (12*a*b*e^4*i^4 - 25*b^2*e^4*i^4 + 12*a*b*f^4*h^4 - 108*b^2*e^2*f^2*h^2* i^2 + 48*b^2*e*f^3*h^3*i + 88*b^2*e^3*f*h*i^3 + 72*a*b*e^2*f^2*h^2*i^2 - 4 8*a*b*e*f^3*h^3*i - 48*a*b*e^3*f*h*i^3)/(12*d*f^5)) - x^2*((e*((i^3*(72*a^ 2*f*h - 7*b^2*e*i + 16*b^2*f*h + 12*a*b*e*i - 48*a*b*f*h))/(18*d*f^2) - (e *i^4*(8*a^2 - 4*a*b + b^2))/(8*d*f^2)))/(2*f) - (i^2*(72*a^2*f^2*h^2 + 13* b^2*e^2*i^2 + 36*b^2*f^2*h^2 - 12*a*b*e^2*i^2 - 72*a*b*f^2*h^2 - 40*b^2*e* f*h*i + 48*a*b*e*f*h*i))/(24*d*f^3)) + x^3*((i^3*(72*a^2*f*h - 7*b^2*e*i + 16*b^2*f*h + 12*a*b*e*i - 48*a*b*f*h))/(54*d*f^2) - (e*i^4*(8*a^2 - 4*a*b + b^2))/(24*d*f^2)) + x*((288*a^2*f^3*h^3*i - 300*b^2*e^3*i^4 + 576*b^2*f ^3*h^3*i + 144*a*b*e^3*i^4 - 576*a*b*f^3*h^3*i + 1056*b^2*e^2*f*h*i^3 - 12 96*b^2*e*f^2*h^2*i^2 - 576*a*b*e^2*f*h*i^3 + 864*a*b*e*f^2*h^2*i^2)/(72*d* f^4) + (e*((e*((i^3*(72*a^2*f*h - 7*b^2*e*i + 16*b^2*f*h + 12*a*b*e*i - 48 *a*b*f*h))/(18*d*f^2) - (e*i^4*(8*a^2 - 4*a*b + b^2))/(8*d*f^2)))/f - (i^2 *(72*a^2*f^2*h^2 + 13*b^2*e^2*i^2 + 36*b^2*f^2*h^2 - 12*a*b*e^2*i^2 - 72*a *b*f^2*h^2 - 40*b^2*e*f*h*i + 48*a*b*e*f*h*i))/(12*d*f^3)))/f) + f*log(c*( e + f*x))*((x^3*(7*b^2*e*i^4 - 12*a*b*e*i^4 - 16*b^2*f*h*i^3 + 48*a*b*f*h* i^3))/(18*d*f^3) - (x^2*(13*b^2*e^2*i^4 + 36*b^2*f^2*h^2*i^2 - 12*a*b*e...