Integrand size = 22, antiderivative size = 400 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=-\frac {77 b^2 e^5 n^2 \sqrt [3]{x}}{60 d^5}+\frac {47 b^2 e^4 n^2 x^{2/3}}{120 d^4}-\frac {3 b^2 e^3 n^2 x}{20 d^3}+\frac {b^2 e^2 n^2 x^{4/3}}{20 d^2}+\frac {77 b^2 e^6 n^2 \log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{60 d^6}+\frac {b e^5 n \left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^6}-\frac {b e^4 n x^{2/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{2 d^4}+\frac {b e^3 n x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{3 d^3}-\frac {b e^2 n x^{4/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{4 d^2}+\frac {b e n x^{5/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{5 d}+\frac {b e^6 n \log \left (1-\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right ) \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d^6}+\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}-\frac {b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+\frac {e}{\sqrt [3]{x}}}\right )}{d^6} \] Output:
-77/60*b^2*e^5*n^2*x^(1/3)/d^5+47/120*b^2*e^4*n^2*x^(2/3)/d^4-3/20*b^2*e^3 *n^2*x/d^3+1/20*b^2*e^2*n^2*x^(4/3)/d^2+77/60*b^2*e^6*n^2*ln(d+e/x^(1/3))/ d^6+b*e^5*n*(d+e/x^(1/3))*x^(1/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d^6-1/2*b*e^ 4*n*x^(2/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d^4+1/3*b*e^3*n*x*(a+b*ln(c*(d+e/x ^(1/3))^n))/d^3-1/4*b*e^2*n*x^(4/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d^2+1/5*b* e*n*x^(5/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d+b*e^6*n*ln(1-d/(d+e/x^(1/3)))*(a +b*ln(c*(d+e/x^(1/3))^n))/d^6+1/2*x^2*(a+b*ln(c*(d+e/x^(1/3))^n))^2+137/18 0*b^2*e^6*n^2*ln(x)/d^6-b^2*e^6*n^2*polylog(2,d/(d+e/x^(1/3)))/d^6
Time = 0.35 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.09 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\frac {1}{2} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2+\frac {b e n \left (360 a d e^4 \sqrt [3]{x}-462 b d e^4 n \sqrt [3]{x}-180 a d^2 e^3 x^{2/3}+141 b d^2 e^3 n x^{2/3}+120 a d^3 e^2 x-54 b d^3 e^2 n x-90 a d^4 e x^{4/3}+18 b d^4 e n x^{4/3}+72 a d^5 x^{5/3}+642 b e^5 n \log \left (d+\frac {e}{\sqrt [3]{x}}\right )+360 b d e^4 \sqrt [3]{x} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-180 b d^2 e^3 x^{2/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+120 b d^3 e^2 x \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-90 b d^4 e x^{4/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )+72 b d^5 x^{5/3} \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )-360 a e^5 \log \left (e+d \sqrt [3]{x}\right )+180 b e^5 n \log \left (e+d \sqrt [3]{x}\right )-360 b e^5 \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right ) \log \left (e+d \sqrt [3]{x}\right )+180 b e^5 n \log ^2\left (e+d \sqrt [3]{x}\right )-360 b e^5 n \log \left (e+d \sqrt [3]{x}\right ) \log \left (-\frac {d \sqrt [3]{x}}{e}\right )+214 b e^5 n \log (x)-360 b e^5 n \operatorname {PolyLog}\left (2,1+\frac {d \sqrt [3]{x}}{e}\right )\right )}{360 d^6} \] Input:
Integrate[x*(a + b*Log[c*(d + e/x^(1/3))^n])^2,x]
Output:
(x^2*(a + b*Log[c*(d + e/x^(1/3))^n])^2)/2 + (b*e*n*(360*a*d*e^4*x^(1/3) - 462*b*d*e^4*n*x^(1/3) - 180*a*d^2*e^3*x^(2/3) + 141*b*d^2*e^3*n*x^(2/3) + 120*a*d^3*e^2*x - 54*b*d^3*e^2*n*x - 90*a*d^4*e*x^(4/3) + 18*b*d^4*e*n*x^ (4/3) + 72*a*d^5*x^(5/3) + 642*b*e^5*n*Log[d + e/x^(1/3)] + 360*b*d*e^4*x^ (1/3)*Log[c*(d + e/x^(1/3))^n] - 180*b*d^2*e^3*x^(2/3)*Log[c*(d + e/x^(1/3 ))^n] + 120*b*d^3*e^2*x*Log[c*(d + e/x^(1/3))^n] - 90*b*d^4*e*x^(4/3)*Log[ c*(d + e/x^(1/3))^n] + 72*b*d^5*x^(5/3)*Log[c*(d + e/x^(1/3))^n] - 360*a*e ^5*Log[e + d*x^(1/3)] + 180*b*e^5*n*Log[e + d*x^(1/3)] - 360*b*e^5*Log[c*( d + e/x^(1/3))^n]*Log[e + d*x^(1/3)] + 180*b*e^5*n*Log[e + d*x^(1/3)]^2 - 360*b*e^5*n*Log[e + d*x^(1/3)]*Log[-((d*x^(1/3))/e)] + 214*b*e^5*n*Log[x] - 360*b*e^5*n*PolyLog[2, 1 + (d*x^(1/3))/e]))/(360*d^6)
Time = 3.75 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.40, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.136, Rules used = {2904, 2845, 2858, 27, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}
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.
| \(\displaystyle \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -3 \int x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle -3 \left (\frac {1}{3} b e n \int \frac {x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )}{d+\frac {e}{\sqrt [3]{x}}}d\frac {1}{\sqrt [3]{x}}-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle -3 \left (\frac {1}{3} b n \int x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \int \frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^6}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^6}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {-\frac {1}{5} b n \int -\frac {x^2}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {-\frac {1}{5} b n \int \left (-\frac {x^{5/3}}{d e^5}+\frac {x^{4/3}}{d^2 e^4}-\frac {x}{d^3 e^3}+\frac {x^{2/3}}{d^4 e^2}-\frac {\sqrt [3]{x}}{d^5 e}+\frac {\sqrt [3]{x}}{d^5}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\int -\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \int \frac {x^{5/3}}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \int \left (\frac {x^{4/3}}{d e^4}-\frac {x}{d^2 e^3}+\frac {x^{2/3}}{d^3 e^2}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {\sqrt [3]{x}}{d^4}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {-\frac {1}{3} b n \int -\frac {x^{4/3}}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {-\frac {1}{3} b n \int \left (-\frac {x}{d e^3}+\frac {x^{2/3}}{d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\sqrt [3]{x}}{d^3}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \frac {x}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \left (\frac {x^{2/3}}{d e^2}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {\sqrt [3]{x}}{d^2}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {\int \frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {-\frac {b n \int -\frac {\sqrt [3]{x}}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}}{d}+\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {\frac {b n \int \sqrt [3]{x} \log \left (1-d \sqrt [3]{x}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}}{d}+\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -3 \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\sqrt [3]{x} \left (d+\frac {e}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}}{d}+\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{5 e^5}-\frac {1}{5} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^5}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {x^{2/3}}{2 d^3 e^2}-\frac {x}{3 d^2 e^3}+\frac {x^{4/3}}{4 d e^4}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2\right )\) |
Input:
Int[x*(a + b*Log[c*(d + e/x^(1/3))^n])^2,x]
Output:
-3*(-1/6*(x^2*(a + b*Log[c*(d + e/x^(1/3))^n])^2) + (b*e^6*n*((-1/5*(b*n*( -(x^(1/3)/(d^4*e)) + x^(2/3)/(2*d^3*e^2) - x/(3*d^2*e^3) + x^(4/3)/(4*d*e^ 4) + Log[d + e/x^(1/3)]/d^5 - Log[-(e/x^(1/3))]/d^5)) - (x^(5/3)*(a + b*Lo g[c/x^(n/3)]))/(5*e^5))/d + ((-1/4*(b*n*(-(x^(1/3)/(d^3*e)) + x^(2/3)/(2*d ^2*e^2) - x/(3*d*e^3) + Log[d + e/x^(1/3)]/d^4 - Log[-(e/x^(1/3))]/d^4)) + (x^(4/3)*(a + b*Log[c/x^(n/3)]))/(4*e^4))/d + ((-1/3*(b*n*(-(x^(1/3)/(d^2 *e)) + x^(2/3)/(2*d*e^2) + Log[d + e/x^(1/3)]/d^3 - Log[-(e/x^(1/3))]/d^3) ) - (x*(a + b*Log[c/x^(n/3)]))/(3*e^3))/d + ((-1/2*(b*n*(-(x^(1/3)/(d*e)) + Log[d + e/x^(1/3)]/d^2 - Log[-(e/x^(1/3))]/d^2)) + (x^(2/3)*(a + b*Log[c /x^(n/3)]))/(2*e^2))/d + (((b*n*Log[-(e/x^(1/3))])/d - ((d + e/x^(1/3))*x^ (1/3)*(a + b*Log[c/x^(n/3)]))/(d*e))/d + (-((Log[1 - d*x^(1/3)]*(a + b*Log [c/x^(n/3)]))/d) + (b*n*PolyLog[2, d*x^(1/3)])/d)/d)/d)/d)/d)/d))/3)
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_))^(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[(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])
\[\int x {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )}^{2}d x\]
Input:
int(x*(a+b*ln(c*(d+e/x^(1/3))^n))^2,x)
Output:
int(x*(a+b*ln(c*(d+e/x^(1/3))^n))^2,x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e/x^(1/3))^n))^2,x, algorithm="fricas")
Output:
integral(b^2*x*log(c*((d*x + e*x^(2/3))/x)^n)^2 + 2*a*b*x*log(c*((d*x + e* x^(2/3))/x)^n) + a^2*x, x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int x \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right )^{2}\, dx \] Input:
integrate(x*(a+b*ln(c*(d+e/x**(1/3))**n))**2,x)
Output:
Integral(x*(a + b*log(c*(d + e/x**(1/3))**n))**2, x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e/x^(1/3))^n))^2,x, algorithm="maxima")
Output:
1/2*b^2*x^2*log((d*x^(1/3) + e)^n)^2 - integrate(-1/3*(3*(b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^2 + 3*(b^2*d*x^2 + b^2*e*x^(5/3))*log(x^(1/3*n) )^2 + 3*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x^(5/3) - (b^2*d*n*x^2 - 6*(b^2*d*log(c) + a*b*d)*x^2 - 6*(b^2*e*log(c) + a*b*e)*x^(5/3) + 6*(b^2* d*x^2 + b^2*e*x^(5/3))*log(x^(1/3*n)))*log((d*x^(1/3) + e)^n) - 6*((b^2*d* log(c) + a*b*d)*x^2 + (b^2*e*log(c) + a*b*e)*x^(5/3))*log(x^(1/3*n)))/(d*x + e*x^(2/3)), x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{2} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e/x^(1/3))^n))^2,x, algorithm="giac")
Output:
integrate((b*log(c*(d + e/x^(1/3))^n) + a)^2*x, x)
Timed out. \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^2 \,d x \] Input:
int(x*(a + b*log(c*(d + e/x^(1/3))^n))^2,x)
Output:
int(x*(a + b*log(c*(d + e/x^(1/3))^n))^2, x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2 \, dx=\frac {24 x^{\frac {5}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right ) b^{2} d^{5} e n -60 x^{\frac {2}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right ) b^{2} d^{2} e^{4} n +24 x^{\frac {5}{3}} a b \,d^{5} e n -60 x^{\frac {2}{3}} a b \,d^{2} e^{4} n +47 x^{\frac {2}{3}} b^{2} d^{2} e^{4} n^{2}+60 x^{\frac {1}{3}} {\mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right )}^{2} b^{2} d \,e^{5}-30 x^{\frac {4}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right ) b^{2} d^{4} e^{2} n +120 x^{\frac {1}{3}} \mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right ) b^{2} d \,e^{5} n -30 x^{\frac {4}{3}} a b \,d^{4} e^{2} n +120 x^{\frac {1}{3}} a b d \,e^{5} n +6 x^{\frac {4}{3}} b^{2} d^{4} e^{2} n^{2}-154 x^{\frac {1}{3}} b^{2} d \,e^{5} n^{2}-20 \left (\int \frac {{\mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right )}^{2}}{x^{\frac {2}{3}}}d x \right ) b^{2} d \,e^{5}-120 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) a b \,e^{6} n +274 \,\mathrm {log}\left (x^{\frac {1}{3}}\right ) b^{2} e^{6} n^{2}+60 {\mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right )}^{2} b^{2} d^{6} x^{2}+120 \,\mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right ) a b \,d^{6} x^{2}-120 \,\mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right ) a b \,e^{6}+40 \,\mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right ) b^{2} d^{3} e^{3} n x +274 \,\mathrm {log}\left (\frac {\left (x^{\frac {1}{3}} d +e \right )^{n} c}{x^{\frac {n}{3}}}\right ) b^{2} e^{6} n +60 a^{2} d^{6} x^{2}+40 a b \,d^{3} e^{3} n x -18 b^{2} d^{3} e^{3} n^{2} x}{120 d^{6}} \] Input:
int(x*(a+b*log(c*(d+e/x^(1/3))^n))^2,x)
Output:
(24*x**(2/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*b**2*d**5*e*n*x - 60*x* *(2/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*b**2*d**2*e**4*n + 24*x**(2/3 )*a*b*d**5*e*n*x - 60*x**(2/3)*a*b*d**2*e**4*n + 47*x**(2/3)*b**2*d**2*e** 4*n**2 + 60*x**(1/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**2*b**2*d*e**5 - 30*x**(1/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*b**2*d**4*e**2*n*x + 1 20*x**(1/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*b**2*d*e**5*n - 30*x**(1 /3)*a*b*d**4*e**2*n*x + 120*x**(1/3)*a*b*d*e**5*n + 6*x**(1/3)*b**2*d**4*e **2*n**2*x - 154*x**(1/3)*b**2*d*e**5*n**2 - 20*int(log(((x**(1/3)*d + e)* *n*c)/x**(n/3))**2/x**(2/3),x)*b**2*d*e**5 - 120*log(x**(1/3))*a*b*e**6*n + 274*log(x**(1/3))*b**2*e**6*n**2 + 60*log(((x**(1/3)*d + e)**n*c)/x**(n/ 3))**2*b**2*d**6*x**2 + 120*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*a*b*d**6 *x**2 - 120*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*a*b*e**6 + 40*log(((x**( 1/3)*d + e)**n*c)/x**(n/3))*b**2*d**3*e**3*n*x + 274*log(((x**(1/3)*d + e) **n*c)/x**(n/3))*b**2*e**6*n + 60*a**2*d**6*x**2 + 40*a*b*d**3*e**3*n*x - 18*b**2*d**3*e**3*n**2*x)/(120*d**6)