Integrand size = 24, antiderivative size = 572 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\frac {481 b^2 e^8 n^2 \sqrt [3]{x}}{420 d^8}-\frac {341 b^2 e^7 n^2 x^{2/3}}{840 d^7}+\frac {743 b^2 e^6 n^2 x}{3780 d^6}-\frac {533 b^2 e^5 n^2 x^{4/3}}{5040 d^5}+\frac {73 b^2 e^4 n^2 x^{5/3}}{1260 d^4}-\frac {5 b^2 e^3 n^2 x^2}{168 d^3}+\frac {b^2 e^2 n^2 x^{7/3}}{84 d^2}-\frac {481 b^2 e^9 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{420 d^9}-\frac {2 b e^8 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac {b e^7 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^7}-\frac {2 b e^6 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^6}+\frac {b e^5 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{6 d^5}-\frac {2 b e^4 n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{15 d^4}+\frac {b e^3 n x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{9 d^3}-\frac {2 b e^2 n x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{21 d^2}+\frac {b e n x^{8/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{12 d}-\frac {2 b e^9 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^9}+\frac {1}{3} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2-\frac {761 b^2 e^9 n^2 \log (x)}{1260 d^9}+\frac {2 b^2 e^9 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{3 d^9} \]
481/420*b^2*e^8*n^2*x^(1/3)/d^8-341/840*b^2*e^7*n^2*x^(2/3)/d^7+743/3780*b ^2*e^6*n^2*x/d^6-533/5040*b^2*e^5*n^2*x^(4/3)/d^5+73/1260*b^2*e^4*n^2*x^(5 /3)/d^4-5/168*b^2*e^3*n^2*x^2/d^3+1/84*b^2*e^2*n^2*x^(7/3)/d^2-481/420*b^2 *e^9*n^2*ln(d+e/x^(1/3))/d^9-2/3*b*e^8*n*(d+e/x^(1/3))*x^(1/3)*(a+b*ln(c*( d+e/x^(1/3))^n))/d^9+1/3*b*e^7*n*x^(2/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d^7-2 /9*b*e^6*n*x*(a+b*ln(c*(d+e/x^(1/3))^n))/d^6+1/6*b*e^5*n*x^(4/3)*(a+b*ln(c *(d+e/x^(1/3))^n))/d^5-2/15*b*e^4*n*x^(5/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d^ 4+1/9*b*e^3*n*x^2*(a+b*ln(c*(d+e/x^(1/3))^n))/d^3-2/21*b*e^2*n*x^(7/3)*(a+ b*ln(c*(d+e/x^(1/3))^n))/d^2+1/12*b*e*n*x^(8/3)*(a+b*ln(c*(d+e/x^(1/3))^n) )/d-2/3*b*e^9*n*ln(1-d/(d+e/x^(1/3)))*(a+b*ln(c*(d+e/x^(1/3))^n))/d^9+1/3* x^3*(a+b*ln(c*(d+e/x^(1/3))^n))^2-761/1260*b^2*e^9*n^2*ln(x)/d^9+2/3*b^2*e ^9*n^2*polylog(2,d/(d+e/x^(1/3)))/d^9
Time = 0.42 (sec) , antiderivative size = 603, normalized size of antiderivative = 1.05 \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\frac {1}{3} \left (x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {b e n \left (-10080 a d e^7 \sqrt [3]{x}+17316 b d e^7 n \sqrt [3]{x}+5040 a d^2 e^6 x^{2/3}-6138 b d^2 e^6 n x^{2/3}-3360 a d^3 e^5 x+2972 b d^3 e^5 n x+2520 a d^4 e^4 x^{4/3}-1599 b d^4 e^4 n x^{4/3}-2016 a d^5 e^3 x^{5/3}+876 b d^5 e^3 n x^{5/3}+1680 a d^6 e^2 x^2-450 b d^6 e^2 n x^2-1440 a d^7 e x^{7/3}+180 b d^7 e n x^{7/3}+1260 a d^8 x^{8/3}-22356 b e^8 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )-10080 b d e^7 \sqrt [3]{x} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+5040 b d^2 e^6 x^{2/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-3360 b d^3 e^5 x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+2520 b d^4 e^4 x^{4/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-2016 b d^5 e^3 x^{5/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+1680 b d^6 e^2 x^2 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-1440 b d^7 e x^{7/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+1260 b d^8 x^{8/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+10080 a e^8 \log \left (e+d \sqrt [3]{x}\right )-5040 b e^8 n \log \left (e+d \sqrt [3]{x}\right )+10080 b e^8 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \log \left (e+d \sqrt [3]{x}\right )-5040 b e^8 n \log ^2\left (e+d \sqrt [3]{x}\right )+10080 b e^8 n \log \left (e+d \sqrt [3]{x}\right ) \log \left (-\frac {d \sqrt [3]{x}}{e}\right )-7452 b e^8 n \log (x)+10080 b e^8 n \operatorname {PolyLog}\left (2,1+\frac {d \sqrt [3]{x}}{e}\right )\right )}{5040 d^9}\right ) \]
(x^3*(a + b*Log[c*(d + e/x^(1/3))^n])^2 + (b*e*n*(-10080*a*d*e^7*x^(1/3) + 17316*b*d*e^7*n*x^(1/3) + 5040*a*d^2*e^6*x^(2/3) - 6138*b*d^2*e^6*n*x^(2/ 3) - 3360*a*d^3*e^5*x + 2972*b*d^3*e^5*n*x + 2520*a*d^4*e^4*x^(4/3) - 1599 *b*d^4*e^4*n*x^(4/3) - 2016*a*d^5*e^3*x^(5/3) + 876*b*d^5*e^3*n*x^(5/3) + 1680*a*d^6*e^2*x^2 - 450*b*d^6*e^2*n*x^2 - 1440*a*d^7*e*x^(7/3) + 180*b*d^ 7*e*n*x^(7/3) + 1260*a*d^8*x^(8/3) - 22356*b*e^8*n*Log[d + e/x^(1/3)] - 10 080*b*d*e^7*x^(1/3)*Log[c*(d + e/x^(1/3))^n] + 5040*b*d^2*e^6*x^(2/3)*Log[ c*(d + e/x^(1/3))^n] - 3360*b*d^3*e^5*x*Log[c*(d + e/x^(1/3))^n] + 2520*b* d^4*e^4*x^(4/3)*Log[c*(d + e/x^(1/3))^n] - 2016*b*d^5*e^3*x^(5/3)*Log[c*(d + e/x^(1/3))^n] + 1680*b*d^6*e^2*x^2*Log[c*(d + e/x^(1/3))^n] - 1440*b*d^ 7*e*x^(7/3)*Log[c*(d + e/x^(1/3))^n] + 1260*b*d^8*x^(8/3)*Log[c*(d + e/x^( 1/3))^n] + 10080*a*e^8*Log[e + d*x^(1/3)] - 5040*b*e^8*n*Log[e + d*x^(1/3) ] + 10080*b*e^8*Log[c*(d + e/x^(1/3))^n]*Log[e + d*x^(1/3)] - 5040*b*e^8*n *Log[e + d*x^(1/3)]^2 + 10080*b*e^8*n*Log[e + d*x^(1/3)]*Log[-((d*x^(1/3)) /e)] - 7452*b*e^8*n*Log[x] + 10080*b*e^8*n*PolyLog[2, 1 + (d*x^(1/3))/e])) /(5040*d^9))/3
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.
| \(\displaystyle \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -3 \int x^{10/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle -3 \left (\frac {2}{9} b e n \int \frac {x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d+\frac {e}{\sqrt [3]{x}}}d\frac {1}{\sqrt [3]{x}}-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle -3 \left (\frac {2}{9} b n \int x^{10/3} \left (a+b \log \left (c x^{-n/3}\right )\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -3 \left (-\frac {2}{9} b n \int -x^{10/3} \left (a+b \log \left (c x^{-n/3}\right )\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \int -\frac {x^{10/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^9}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\int -\frac {x^3 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^9}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^3 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^8}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \int \frac {x^3}{e^8}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^3 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^8}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \int \left (\frac {x^{8/3}}{d e^8}-\frac {x^{7/3}}{d^2 e^7}+\frac {x^2}{d^3 e^6}-\frac {x^{5/3}}{d^4 e^5}+\frac {x^{4/3}}{d^5 e^4}-\frac {x}{d^6 e^3}+\frac {x^{2/3}}{d^7 e^2}-\frac {\sqrt [3]{x}}{d^8 e}+\frac {\sqrt [3]{x}}{d^8}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^3 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^8}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\int \frac {x^3 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^8}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\int \frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^8}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^7}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {-\frac {1}{7} b n \int -\frac {x^{8/3}}{e^7}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}}{d}+\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^7}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {-\frac {1}{7} b n \int \left (-\frac {x^{7/3}}{d e^7}+\frac {x^2}{d^2 e^6}-\frac {x^{5/3}}{d^3 e^5}+\frac {x^{4/3}}{d^4 e^4}-\frac {x}{d^5 e^3}+\frac {x^{2/3}}{d^6 e^2}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {\sqrt [3]{x}}{d^7}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}}{d}+\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^7}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\int -\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^7}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\int -\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^7}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^6}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\int \frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^6}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \int \frac {x^{7/3}}{e^6}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \int \left (\frac {x^2}{d e^6}-\frac {x^{5/3}}{d^2 e^5}+\frac {x^{4/3}}{d^3 e^4}-\frac {x}{d^4 e^3}+\frac {x^{2/3}}{d^5 e^2}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {\sqrt [3]{x}}{d^6}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^6}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\int \frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^6}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {x}{3 d^3 e^3}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x^{5/3}}{5 d e^5}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^6}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {x}{3 d^3 e^3}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x^{5/3}}{5 d e^5}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\frac {-\frac {1}{5} b n \int -\frac {x^2}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {x}{3 d^3 e^3}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x^{5/3}}{5 d e^5}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\frac {-\frac {1}{5} b n \int \left (-\frac {x^{5/3}}{d e^5}+\frac {x^{4/3}}{d^2 e^4}-\frac {x}{d^3 e^3}+\frac {x^{2/3}}{d^4 e^2}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {\sqrt [3]{x}}{d^5}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {x}{3 d^3 e^3}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x^{5/3}}{5 d e^5}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {x}{3 d^3 e^3}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x^{5/3}}{5 d e^5}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\frac {\frac {\int -\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {x}{3 d^3 e^3}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x^{5/3}}{5 d e^5}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \int \frac {x^{5/3}}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {x}{3 d^3 e^3}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x^{5/3}}{5 d e^5}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \int \left (\frac {x^{4/3}}{d e^4}-\frac {x}{d^2 e^3}+\frac {x^{2/3}}{d^3 e^2}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {\sqrt [3]{x}}{d^4}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {x}{3 d^3 e^3}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x^{5/3}}{5 d e^5}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (-\frac {2}{9} b e^9 n \left (\frac {\frac {\frac {\frac {\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}}{d}+\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {x}{3 d^3 e^3}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x^{5/3}}{5 d e^5}\right )}{d}}{d}+\frac {-\frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{7 e^7}-\frac {1}{7} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {x}{3 d^4 e^3}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^2}{6 d e^6}\right )}{d}}{d}+\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {x}{3 d^5 e^3}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^2}{6 d^2 e^6}-\frac {x^{7/3}}{7 d e^7}\right )}{d}\right )-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (-\frac {2}{9} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (-\frac {x^{7/3}}{7 d e^7}+\frac {x^2}{6 d^2 e^6}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x}{3 d^5 e^3}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-n/3}\right )\right ) x^{7/3}}{7 e^7}-\frac {1}{7} b n \left (\frac {x^2}{6 d e^6}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x}{3 d^4 e^3}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}\right )}{d}+\frac {\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (-\frac {x^{5/3}}{5 d e^5}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x}{3 d^3 e^3}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-n/3}\right )\right ) x^{5/3}}{5 e^5}-\frac {1}{5} b n \left (\frac {x^{4/3}}{4 d e^4}-\frac {x}{3 d^2 e^3}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}}{d}}{d}\right ) e^9-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (-\frac {2}{9} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (-\frac {x^{7/3}}{7 d e^7}+\frac {x^2}{6 d^2 e^6}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x}{3 d^5 e^3}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-n/3}\right )\right ) x^{7/3}}{7 e^7}-\frac {1}{7} b n \left (\frac {x^2}{6 d e^6}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x}{3 d^4 e^3}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}\right )}{d}+\frac {\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (-\frac {x^{5/3}}{5 d e^5}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x}{3 d^3 e^3}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-n/3}\right )\right ) x^{5/3}}{5 e^5}-\frac {1}{5} b n \left (\frac {x^{4/3}}{4 d e^4}-\frac {x}{3 d^2 e^3}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \int -\frac {x^{4/3}}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}}{d}}{d}\right ) e^9-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (-\frac {2}{9} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (-\frac {x^{7/3}}{7 d e^7}+\frac {x^2}{6 d^2 e^6}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x}{3 d^5 e^3}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-n/3}\right )\right ) x^{7/3}}{7 e^7}-\frac {1}{7} b n \left (\frac {x^2}{6 d e^6}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x}{3 d^4 e^3}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}\right )}{d}+\frac {\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (-\frac {x^{5/3}}{5 d e^5}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x}{3 d^3 e^3}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-n/3}\right )\right ) x^{5/3}}{5 e^5}-\frac {1}{5} b n \left (\frac {x^{4/3}}{4 d e^4}-\frac {x}{3 d^2 e^3}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \int \left (-\frac {x}{d e^3}+\frac {x^{2/3}}{d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\sqrt [3]{x}}{d^3}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}}{d}}{d}\right ) e^9-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (-\frac {2}{9} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (-\frac {x^{7/3}}{7 d e^7}+\frac {x^2}{6 d^2 e^6}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x}{3 d^5 e^3}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-n/3}\right )\right ) x^{7/3}}{7 e^7}-\frac {1}{7} b n \left (\frac {x^2}{6 d e^6}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x}{3 d^4 e^3}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}\right )}{d}+\frac {\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (-\frac {x^{5/3}}{5 d e^5}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x}{3 d^3 e^3}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-n/3}\right )\right ) x^{5/3}}{5 e^5}-\frac {1}{5} b n \left (\frac {x^{4/3}}{4 d e^4}-\frac {x}{3 d^2 e^3}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}}{d}}{d}\right ) e^9-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (-\frac {2}{9} b n \left (\frac {\frac {x^{8/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{8 e^8}-\frac {1}{8} b n \left (-\frac {x^{7/3}}{7 d e^7}+\frac {x^2}{6 d^2 e^6}-\frac {x^{5/3}}{5 d^3 e^5}+\frac {x^{4/3}}{4 d^4 e^4}-\frac {x}{3 d^5 e^3}+\frac {x^{2/3}}{2 d^6 e^2}-\frac {\sqrt [3]{x}}{d^7 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^8}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^8}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-n/3}\right )\right ) x^{7/3}}{7 e^7}-\frac {1}{7} b n \left (\frac {x^2}{6 d e^6}-\frac {x^{5/3}}{5 d^2 e^5}+\frac {x^{4/3}}{4 d^3 e^4}-\frac {x}{3 d^4 e^3}+\frac {x^{2/3}}{2 d^5 e^2}-\frac {\sqrt [3]{x}}{d^6 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^7}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^7}\right )}{d}+\frac {\frac {\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{6 e^6}-\frac {1}{6} b n \left (-\frac {x^{5/3}}{5 d e^5}+\frac {x^{4/3}}{4 d^2 e^4}-\frac {x}{3 d^3 e^3}+\frac {x^{2/3}}{2 d^4 e^2}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^6}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^6}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{-n/3}\right )\right ) x^{5/3}}{5 e^5}-\frac {1}{5} b n \left (\frac {x^{4/3}}{4 d e^4}-\frac {x}{3 d^2 e^3}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}}{d}}{d}}{d}\right ) e^9-\frac {1}{9} x^3 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
3.5.97.3.1 Defintions of rubi rules used
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_.))^(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_))^(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[((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])
\[\int x^{2} {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )}^{2}d x\]
\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \]
integral(b^2*x^2*log(c*((d*x + e*x^(2/3))/x)^n)^2 + 2*a*b*x^2*log(c*((d*x + e*x^(2/3))/x)^n) + a^2*x^2, x)
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]
\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \]
1/3*b^2*x^3*log((d*x^(1/3) + e)^n)^2 - integrate(-1/9*(9*(b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^3 + 9*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e) *x^(8/3) + 9*(b^2*d*x^3 + b^2*e*x^(8/3))*log(x^(1/3*n))^2 - 2*(b^2*d*n*x^3 - 9*(b^2*d*log(c) + a*b*d)*x^3 - 9*(b^2*e*log(c) + a*b*e)*x^(8/3) + 9*(b^ 2*d*x^3 + b^2*e*x^(8/3))*log(x^(1/3*n)))*log((d*x^(1/3) + e)^n) - 18*((b^2 *d*log(c) + a*b*d)*x^3 + (b^2*e*log(c) + a*b*e)*x^(8/3))*log(x^(1/3*n)))/( d*x + e*x^(2/3)), x)
\[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int x^2\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^2 \,d x \]