Integrand size = 24, antiderivative size = 773 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\frac {71 b^3 e^5 n^3 x^{2/3}}{80 d^5}-\frac {3 b^3 e^4 n^3 x^{4/3}}{20 d^4}+\frac {b^3 e^3 n^3 x^2}{40 d^3}-\frac {71 b^3 e^6 n^3 \log \left (d+\frac {e}{x^{2/3}}\right )}{80 d^6}-\frac {77 b^2 e^5 n^2 \left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^6}+\frac {47 b^2 e^4 n^2 x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{80 d^4}-\frac {9 b^2 e^3 n^2 x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^3}+\frac {3 b^2 e^2 n^2 x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^2}-\frac {77 b^2 e^6 n^2 \log \left (1-\frac {d}{d+\frac {e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )}{40 d^6}+\frac {3 b e^5 n \left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 d^6}-\frac {3 b e^4 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{8 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 d^3}-\frac {3 b e^2 n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{16 d^2}+\frac {3 b e n x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{20 d}+\frac {3 b e^6 n \log \left (1-\frac {d}{d+\frac {e}{x^{2/3}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{4 d^6}+\frac {1}{4} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3-\frac {3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \log \left (-\frac {e}{d x^{2/3}}\right )}{2 d^6}-\frac {15 b^3 e^6 n^3 \log (x)}{8 d^6}+\frac {77 b^3 e^6 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{40 d^6}-\frac {3 b^2 e^6 n^2 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{2 d^6}-\frac {3 b^3 e^6 n^3 \operatorname {PolyLog}\left (2,1+\frac {e}{d x^{2/3}}\right )}{2 d^6}-\frac {3 b^3 e^6 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+\frac {e}{x^{2/3}}}\right )}{2 d^6} \]
71/80*b^3*e^5*n^3*x^(2/3)/d^5-3/20*b^3*e^4*n^3*x^(4/3)/d^4+1/40*b^3*e^3*n^ 3*x^2/d^3-71/80*b^3*e^6*n^3*ln(d+e/x^(2/3))/d^6-77/40*b^2*e^5*n^2*(d+e/x^( 2/3))*x^(2/3)*(a+b*ln(c*(d+e/x^(2/3))^n))/d^6+47/80*b^2*e^4*n^2*x^(4/3)*(a +b*ln(c*(d+e/x^(2/3))^n))/d^4-9/40*b^2*e^3*n^2*x^2*(a+b*ln(c*(d+e/x^(2/3)) ^n))/d^3+3/40*b^2*e^2*n^2*x^(8/3)*(a+b*ln(c*(d+e/x^(2/3))^n))/d^2-77/40*b^ 2*e^6*n^2*ln(1-d/(d+e/x^(2/3)))*(a+b*ln(c*(d+e/x^(2/3))^n))/d^6+3/4*b*e^5* n*(d+e/x^(2/3))*x^(2/3)*(a+b*ln(c*(d+e/x^(2/3))^n))^2/d^6-3/8*b*e^4*n*x^(4 /3)*(a+b*ln(c*(d+e/x^(2/3))^n))^2/d^4+1/4*b*e^3*n*x^2*(a+b*ln(c*(d+e/x^(2/ 3))^n))^2/d^3-3/16*b*e^2*n*x^(8/3)*(a+b*ln(c*(d+e/x^(2/3))^n))^2/d^2+3/20* b*e*n*x^(10/3)*(a+b*ln(c*(d+e/x^(2/3))^n))^2/d+3/4*b*e^6*n*ln(1-d/(d+e/x^( 2/3)))*(a+b*ln(c*(d+e/x^(2/3))^n))^2/d^6+1/4*x^4*(a+b*ln(c*(d+e/x^(2/3))^n ))^3-3/2*b^2*e^6*n^2*(a+b*ln(c*(d+e/x^(2/3))^n))*ln(-e/d/x^(2/3))/d^6-15/8 *b^3*e^6*n^3*ln(x)/d^6+77/40*b^3*e^6*n^3*polylog(2,d/(d+e/x^(2/3)))/d^6-3/ 2*b^2*e^6*n^2*(a+b*ln(c*(d+e/x^(2/3))^n))*polylog(2,d/(d+e/x^(2/3)))/d^6-3 /2*b^3*e^6*n^3*polylog(2,1+e/d/x^(2/3))/d^6-3/2*b^3*e^6*n^3*polylog(3,d/(d +e/x^(2/3)))/d^6
Result contains complex when optimal does not.
Time = 18.53 (sec) , antiderivative size = 5557, normalized size of antiderivative = 7.19 \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\text {Result too large to show} \]
Time = 6.08 (sec) , antiderivative size = 1384, normalized size of antiderivative = 1.79, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {2904, 2845, 2858, 27, 2789, 2756, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2751, 16, 2755, 2754, 2779, 2821, 2838, 7143}
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.
| \(\displaystyle \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -\frac {3}{2} \int x^{14/3} \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3d\frac {1}{x^{2/3}}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e n \int \frac {x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^2}{d+\frac {e}{x^{2/3}}}d\frac {1}{x^{2/3}}-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b n \int x^{14/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2d\left (d+\frac {e}{x^{2/3}}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \int \frac {x^{14/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^6}d\left (d+\frac {e}{x^{2/3}}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {\int \frac {x^4 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^6}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^4 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^5}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \int -\frac {x^4 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^5}d\left (d+\frac {e}{x^{2/3}}\right )-\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\int -\frac {x^4 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^5}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\int -\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^5}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )-\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\frac {\int -\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^5}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \int \frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \int \frac {x^{10/3}}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )-\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{5 e^5}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \int \left (\frac {x^{8/3}}{d e^4}-\frac {x^2}{d^2 e^3}+\frac {x^{4/3}}{d^3 e^2}-\frac {x^{2/3}}{d^4 e}+\frac {x^{2/3}}{d^4}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )-\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \int \frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\int \frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}-\frac {x^{2/3}}{d^3 e}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^2}{3 d e^3}\right )}{d}\right )-\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \int \frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\frac {\int \frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}-\frac {x^{2/3}}{d^3 e}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^2}{3 d e^3}\right )}{d}\right )-\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {\int \frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^4}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {-\frac {1}{3} b n \int -\frac {x^{8/3}}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{3 e^3}}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}-\frac {x^{2/3}}{d^3 e}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^2}{3 d e^3}\right )}{d}\right )-\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\frac {\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {-\frac {2}{3} b n \int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {-\frac {1}{3} b n \int -\frac {x^{8/3}}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{3 e^3}}{d}\right )}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \int \left (-\frac {x^2}{d e^3}+\frac {x^{4/3}}{d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {x^{2/3}}{d^3}\right )d\left (d+\frac {e}{x^{2/3}}\right )-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {-\frac {2}{3} b n \int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}}{d}}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {-\frac {1}{3} b n \int \left (-\frac {x^2}{d e^3}+\frac {x^{4/3}}{d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {x^{2/3}}{d^3}\right )d\left (d+\frac {e}{x^{2/3}}\right )-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}-\frac {x^{2/3}}{d^3 e}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^2}{3 d e^3}\right )}{d}\right )-\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{5 e^5}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}-\frac {x^{2/3}}{d^2 e}+\frac {x^{4/3}}{2 d e^2}\right )}{d}\right )}{d}+\frac {\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {-\frac {2}{3} b n \int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}}{d}}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}-\frac {x^{2/3}}{d^2 e}+\frac {x^{4/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}-\frac {x^{2/3}}{d^3 e}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^2}{3 d e^3}\right )}{d}\right )-\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{5 e^5}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}-\frac {x^{2/3}}{d^2 e}+\frac {x^{4/3}}{2 d e^2}\right )}{d}\right )}{d}+\frac {\frac {-\frac {2}{3} b n \left (\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}}{d}+\frac {\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^3}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}-\frac {x^{2/3}}{d^2 e}+\frac {x^{4/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}-\frac {x^{2/3}}{d^3 e}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^2}{3 d e^3}\right )}{d}\right )-\frac {x^{10/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{5 e^5}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \frac {x^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \frac {x^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \frac {x^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \left (\frac {x^{4/3}}{d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {x^{2/3}}{d^2}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \left (\frac {x^{4/3}}{d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {x^{2/3}}{d^2}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \left (\frac {x^{4/3}}{d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {x^{2/3}}{d^2}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}-\frac {b n \int -\frac {x^{2/3}}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}-\frac {b n \int -\frac {x^{2/3}}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}-\frac {b n \int -\frac {x^{2/3}}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}-\frac {b n \int -\frac {x^{2/3}}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{x^{2/3}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2755 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}-\frac {2 b n \int -\frac {x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}-\frac {2 b n \left (b n \int x^{2/3} \log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right )d\left (d+\frac {e}{x^{2/3}}\right )-\log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )\right )}{d}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{e}d\left (d+\frac {e}{x^{2/3}}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int x^{2/3} \log \left (1-d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int x^{2/3} \log \left (1-d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int x^{2/3} \log \left (1-d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int x^{2/3} \log \left (1-d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}-\frac {2 b n \left (b n \int x^{2/3} \log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right )d\left (d+\frac {e}{x^{2/3}}\right )-\log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )\right )}{d}}{d}+\frac {\frac {2 b n \int x^{2/3} \log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d}}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int x^{2/3} \log \left (1-d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int x^{2/3} \log \left (1-d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int x^{2/3} \log \left (1-d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int x^{2/3} \log \left (1-d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}-\frac {2 b n \left (b n \int x^{2/3} \log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right )d\left (d+\frac {e}{x^{2/3}}\right )-\log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )\right )}{d}}{d}+\frac {\frac {2 b n \left (\left (a+b \log \left (c x^{-2 n/3}\right )\right ) \operatorname {PolyLog}\left (2,d x^{2/3}\right )-b n \int x^{2/3} \operatorname {PolyLog}\left (2,d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d}}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d x^{2/3}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d x^{2/3}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d x^{2/3}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d x^{2/3}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}-\frac {2 b n \left (-\log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{x^{2/3}}}{d}\right )\right )}{d}}{d}+\frac {\frac {2 b n \left (\left (a+b \log \left (c x^{-2 n/3}\right )\right ) \operatorname {PolyLog}\left (2,d x^{2/3}\right )-b n \int x^{2/3} \operatorname {PolyLog}\left (2,d x^{2/3}\right )d\left (d+\frac {e}{x^{2/3}}\right )\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d}}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {3}{2} \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right )^2 x^{10/3}}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x^2}{3 d e^3}+\frac {x^{4/3}}{2 d^2 e^2}-\frac {x^{2/3}}{d^3 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^4}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d x^{2/3}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {\left (a+b \log \left (c x^{-2 n/3}\right )\right ) x^2}{3 e^3}-\frac {1}{3} b n \left (\frac {x^{4/3}}{2 d e^2}-\frac {x^{2/3}}{d^2 e}+\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^3}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^3}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d x^{2/3}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x^2 \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{x^{2/3}}\right )}{d^2}-\frac {\log \left (-\frac {e}{x^{2/3}}\right )}{d^2}-\frac {x^{2/3}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d x^{2/3}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{x^{2/3}}\right )}{d}-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d x^{2/3}\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{x^{2/3}}\right ) x^{2/3} \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d e}-\frac {2 b n \left (-\log \left (1-\frac {d+\frac {e}{x^{2/3}}}{d}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{x^{2/3}}}{d}\right )\right )}{d}}{d}+\frac {\frac {2 b n \left (\left (a+b \log \left (c x^{-2 n/3}\right )\right ) \operatorname {PolyLog}\left (2,d x^{2/3}\right )+b n \operatorname {PolyLog}\left (3,d x^{2/3}\right )\right )}{d}-\frac {\log \left (1-d x^{2/3}\right ) \left (a+b \log \left (c x^{-2 n/3}\right )\right )^2}{d}}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^4 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3\right )\) |
(-3*(-1/6*(x^4*(a + b*Log[c*(d + e/x^(2/3))^n])^3) + (b*e^6*n*((-1/5*(x^(1 0/3)*(a + b*Log[c/x^((2*n)/3)])^2)/e^5 - (2*b*n*((-1/4*(b*n*(-(x^(2/3)/(d^ 3*e)) + x^(4/3)/(2*d^2*e^2) - x^2/(3*d*e^3) + Log[d + e/x^(2/3)]/d^4 - Log [-(e/x^(2/3))]/d^4)) + (x^(8/3)*(a + b*Log[c/x^((2*n)/3)]))/(4*e^4))/d + ( (-1/3*(b*n*(-(x^(2/3)/(d^2*e)) + x^(4/3)/(2*d*e^2) + Log[d + e/x^(2/3)]/d^ 3 - Log[-(e/x^(2/3))]/d^3)) - (x^2*(a + b*Log[c/x^((2*n)/3)]))/(3*e^3))/d + ((-1/2*(b*n*(-(x^(2/3)/(d*e)) + Log[d + e/x^(2/3)]/d^2 - Log[-(e/x^(2/3) )]/d^2)) + (x^(4/3)*(a + b*Log[c/x^((2*n)/3)]))/(2*e^2))/d + (((b*n*Log[-( e/x^(2/3))])/d - ((d + e/x^(2/3))*x^(2/3)*(a + b*Log[c/x^((2*n)/3)]))/(d*e ))/d + (-((Log[1 - d*x^(2/3)]*(a + b*Log[c/x^((2*n)/3)]))/d) + (b*n*PolyLo g[2, d*x^(2/3)])/d)/d)/d)/d)/d))/5)/d + (((x^(8/3)*(a + b*Log[c/x^((2*n)/3 )])^2)/(4*e^4) - (b*n*((-1/3*(b*n*(-(x^(2/3)/(d^2*e)) + x^(4/3)/(2*d*e^2) + Log[d + e/x^(2/3)]/d^3 - Log[-(e/x^(2/3))]/d^3)) - (x^2*(a + b*Log[c/x^( (2*n)/3)]))/(3*e^3))/d + ((-1/2*(b*n*(-(x^(2/3)/(d*e)) + Log[d + e/x^(2/3) ]/d^2 - Log[-(e/x^(2/3))]/d^2)) + (x^(4/3)*(a + b*Log[c/x^((2*n)/3)]))/(2* e^2))/d + (((b*n*Log[-(e/x^(2/3))])/d - ((d + e/x^(2/3))*x^(2/3)*(a + b*Lo g[c/x^((2*n)/3)]))/(d*e))/d + (-((Log[1 - d*x^(2/3)]*(a + b*Log[c/x^((2*n) /3)]))/d) + (b*n*PolyLog[2, d*x^(2/3)])/d)/d)/d)/d))/2)/d + ((-1/3*(x^2*(a + b*Log[c/x^((2*n)/3)])^2)/e^3 - (2*b*n*((-1/2*(b*n*(-(x^(2/3)/(d*e)) + L og[d + e/x^(2/3)]/d^2 - Log[-(e/x^(2/3))]/d^2)) + (x^(4/3)*(a + b*Log[c...
3.6.24.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 0]
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_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int x^{3} {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {2}{3}}}\right )^{n}\right )\right )}^{3}d x\]
\[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} x^{3} \,d x } \]
integral(b^3*x^3*log(c*((d*x + e*x^(1/3))/x)^n)^3 + 3*a*b^2*x^3*log(c*((d* x + e*x^(1/3))/x)^n)^2 + 3*a^2*b*x^3*log(c*((d*x + e*x^(1/3))/x)^n) + a^3* x^3, x)
Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\text {Timed out} \]
\[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} x^{3} \,d x } \]
1/4*b^3*x^4*log((d*x^(2/3) + e)^n)^3 - integrate(-1/2*(2*(b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*x^4 + 2*(b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c) + a^3*e)*x^(10/3) - 16*(b^3*d*x^4 + b^3*e*x^(10/3))*log(x^(1/3*n))^3 - (b^3*d*n*x^4 - 6*(b^3*d*log(c) + a*b^2* d)*x^4 - 6*(b^3*e*log(c) + a*b^2*e)*x^(10/3) + 12*(b^3*d*x^4 + b^3*e*x^(10 /3))*log(x^(1/3*n)))*log((d*x^(2/3) + e)^n)^2 + 24*((b^3*d*log(c) + a*b^2* d)*x^4 + (b^3*e*log(c) + a*b^2*e)*x^(10/3))*log(x^(1/3*n))^2 + 6*((b^3*d*l og(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^4 + (b^3*e*log(c)^2 + 2*a*b^2*e*lo g(c) + a^2*b*e)*x^(10/3) + 4*(b^3*d*x^4 + b^3*e*x^(10/3))*log(x^(1/3*n))^2 - 4*((b^3*d*log(c) + a*b^2*d)*x^4 + (b^3*e*log(c) + a*b^2*e)*x^(10/3))*lo g(x^(1/3*n)))*log((d*x^(2/3) + e)^n) - 12*((b^3*d*log(c)^2 + 2*a*b^2*d*log (c) + a^2*b*d)*x^4 + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(10/3 ))*log(x^(1/3*n)))/(d*x + e*x^(1/3)), x)
\[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {2}{3}}}\right )}^{n}\right ) + a\right )}^{3} x^{3} \,d x } \]
Timed out. \[ \int x^3 \left (a+b \log \left (c \left (d+\frac {e}{x^{2/3}}\right )^n\right )\right )^3 \, dx=\int x^3\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{2/3}}\right )}^n\right )\right )}^3 \,d x \]