Integrand size = 24, antiderivative size = 451 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^3} \, dx=-\frac {3 b^2 e^2 n^2 \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3 x^{2/3}}-\frac {3 b^2 e^3 n^2 \log \left (1-\frac {d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^3}-\frac {3 b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 d x^{4/3}}+\frac {3 b e^2 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3 x^{2/3}}+\frac {3 b e^3 n \log \left (1-\frac {d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{2 d^3}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{2 x^2}-\frac {3 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \log \left (-\frac {e x^{2/3}}{d}\right )}{d^3}+\frac {b^3 e^3 n^3 \log (x)}{d^3}+\frac {3 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,\frac {d}{d+e x^{2/3}}\right )}{2 d^3}-\frac {3 b^2 e^3 n^2 \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {d}{d+e x^{2/3}}\right )}{d^3}-\frac {3 b^3 e^3 n^3 \operatorname {PolyLog}\left (2,1+\frac {e x^{2/3}}{d}\right )}{d^3}-\frac {3 b^3 e^3 n^3 \operatorname {PolyLog}\left (3,\frac {d}{d+e x^{2/3}}\right )}{d^3} \]
-3/2*b^2*e^2*n^2*(d+e*x^(2/3))*(a+b*ln(c*(d+e*x^(2/3))^n))/d^3/x^(2/3)-3/2 *b^2*e^3*n^2*ln(1-d/(d+e*x^(2/3)))*(a+b*ln(c*(d+e*x^(2/3))^n))/d^3-3/4*b*e *n*(a+b*ln(c*(d+e*x^(2/3))^n))^2/d/x^(4/3)+3/2*b*e^2*n*(d+e*x^(2/3))*(a+b* ln(c*(d+e*x^(2/3))^n))^2/d^3/x^(2/3)+3/2*b*e^3*n*ln(1-d/(d+e*x^(2/3)))*(a+ b*ln(c*(d+e*x^(2/3))^n))^2/d^3-1/2*(a+b*ln(c*(d+e*x^(2/3))^n))^3/x^2-3*b^2 *e^3*n^2*(a+b*ln(c*(d+e*x^(2/3))^n))*ln(-e*x^(2/3)/d)/d^3+b^3*e^3*n^3*ln(x )/d^3+3/2*b^3*e^3*n^3*polylog(2,d/(d+e*x^(2/3)))/d^3-3*b^2*e^3*n^2*(a+b*ln (c*(d+e*x^(2/3))^n))*polylog(2,d/(d+e*x^(2/3)))/d^3-3*b^3*e^3*n^3*polylog( 2,1+e*x^(2/3)/d)/d^3-3*b^3*e^3*n^3*polylog(3,d/(d+e*x^(2/3)))/d^3
Time = 0.55 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.69 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^3} \, dx=\frac {-3 b d^2 e n x^{2/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2+6 b d e^2 n x^{4/3} \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-6 b d^3 n \log \left (d+e x^{2/3}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-6 b e^3 n x^2 \log \left (d+e x^{2/3}\right ) \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2-2 d^3 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3+4 b e^3 n x^2 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2 \log (x)-6 b^2 n^2 \left (a-b n \log \left (d+e x^{2/3}\right )+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right ) \left (\left (d^3+e^3 x^2\right ) \log ^2\left (d+e x^{2/3}\right )+e^2 x^{4/3} \left (d+3 e x^{2/3} \log \left (-\frac {e x^{2/3}}{d}\right )\right )+\log \left (d+e x^{2/3}\right ) \left (d^2 e x^{2/3}-2 d e^2 x^{4/3}-3 e^3 x^2-2 e^3 x^2 \log \left (-\frac {e x^{2/3}}{d}\right )\right )-2 e^3 x^2 \operatorname {PolyLog}\left (2,1+\frac {e x^{2/3}}{d}\right )\right )+b^3 n^3 \left (-6 d e^2 x^{4/3} \log \left (d+e x^{2/3}\right )-6 e^3 x^2 \log \left (d+e x^{2/3}\right )-3 d^2 e x^{2/3} \log ^2\left (d+e x^{2/3}\right )+6 d e^2 x^{4/3} \log ^2\left (d+e x^{2/3}\right )+9 e^3 x^2 \log ^2\left (d+e x^{2/3}\right )-2 d^3 \log ^3\left (d+e x^{2/3}\right )-2 e^3 x^2 \log ^3\left (d+e x^{2/3}\right )+6 e^3 x^2 \log \left (-\frac {e x^{2/3}}{d}\right )-18 e^3 x^2 \log \left (d+e x^{2/3}\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+6 e^3 x^2 \log ^2\left (d+e x^{2/3}\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+6 e^3 x^2 \left (-3+2 \log \left (d+e x^{2/3}\right )\right ) \operatorname {PolyLog}\left (2,1+\frac {e x^{2/3}}{d}\right )-12 e^3 x^2 \operatorname {PolyLog}\left (3,1+\frac {e x^{2/3}}{d}\right )\right )}{4 d^3 x^2} \]
(-3*b*d^2*e*n*x^(2/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3) )^n])^2 + 6*b*d*e^2*n*x^(4/3)*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e *x^(2/3))^n])^2 - 6*b*d^3*n*Log[d + e*x^(2/3)]*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 - 6*b*e^3*n*x^2*Log[d + e*x^(2/3)]*(a - b *n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2 - 2*d^3*(a - b*n*Log [d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^3 + 4*b*e^3*n*x^2*(a - b*n*L og[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])^2*Log[x] - 6*b^2*n^2*(a - b*n*Log[d + e*x^(2/3)] + b*Log[c*(d + e*x^(2/3))^n])*((d^3 + e^3*x^2)*Log[ d + e*x^(2/3)]^2 + e^2*x^(4/3)*(d + 3*e*x^(2/3)*Log[-((e*x^(2/3))/d)]) + L og[d + e*x^(2/3)]*(d^2*e*x^(2/3) - 2*d*e^2*x^(4/3) - 3*e^3*x^2 - 2*e^3*x^2 *Log[-((e*x^(2/3))/d)]) - 2*e^3*x^2*PolyLog[2, 1 + (e*x^(2/3))/d]) + b^3*n ^3*(-6*d*e^2*x^(4/3)*Log[d + e*x^(2/3)] - 6*e^3*x^2*Log[d + e*x^(2/3)] - 3 *d^2*e*x^(2/3)*Log[d + e*x^(2/3)]^2 + 6*d*e^2*x^(4/3)*Log[d + e*x^(2/3)]^2 + 9*e^3*x^2*Log[d + e*x^(2/3)]^2 - 2*d^3*Log[d + e*x^(2/3)]^3 - 2*e^3*x^2 *Log[d + e*x^(2/3)]^3 + 6*e^3*x^2*Log[-((e*x^(2/3))/d)] - 18*e^3*x^2*Log[d + e*x^(2/3)]*Log[-((e*x^(2/3))/d)] + 6*e^3*x^2*Log[d + e*x^(2/3)]^2*Log[- ((e*x^(2/3))/d)] + 6*e^3*x^2*(-3 + 2*Log[d + e*x^(2/3)])*PolyLog[2, 1 + (e *x^(2/3))/d] - 12*e^3*x^2*PolyLog[3, 1 + (e*x^(2/3))/d]))/(4*d^3*x^2)
Time = 1.92 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.83, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {2904, 2845, 2858, 25, 27, 2789, 2756, 2789, 2751, 16, 2755, 2754, 2779, 2821, 2838, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^3} \, dx\) |
\(\Big \downarrow \) 2904 |
\(\displaystyle \frac {3}{2} \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^{8/3}}dx^{2/3}\) |
\(\Big \downarrow \) 2845 |
\(\displaystyle \frac {3}{2} \left (b e n \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{\left (d+e x^{2/3}\right ) x^2}dx^{2/3}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 2858 |
\(\displaystyle \frac {3}{2} \left (b n \int \frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{x^{8/3}}d\left (d+e x^{2/3}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3}{2} \left (-b n \int -\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{x^{8/3}}d\left (d+e x^{2/3}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \int -\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e^3 x^{8/3}}d\left (d+e x^{2/3}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 2789 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \left (\frac {\int -\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e^3 x^2}d\left (d+e x^{2/3}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e^2 x^2}d\left (d+e x^{2/3}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 2756 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \left (\frac {\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{2 e^2 x^{4/3}}-b n \int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^2 x^2}d\left (d+e x^{2/3}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e^2 x^2}d\left (d+e x^{2/3}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 2789 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \left (\frac {\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{2 e^2 x^{4/3}}-b n \left (\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^2 x^{4/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e^2 x^{4/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 2751 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \left (\frac {\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{2 e^2 x^{4/3}}-b n \left (\frac {-\frac {b n \int -\frac {1}{e x^{2/3}}d\left (d+e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e^2 x^{4/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \left (\frac {\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{2 e^2 x^{4/3}}-b n \left (\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\frac {b n \log \left (-e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}\right )}{d}+\frac {\frac {\int \frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e^2 x^{4/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 2755 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \left (\frac {\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{2 e^2 x^{4/3}}-b n \left (\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\frac {b n \log \left (-e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}\right )}{d}+\frac {\frac {-\frac {2 b n \int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e x^{2/3}}d\left (d+e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{d e x^{2/3}}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 2754 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \left (\frac {\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{2 e^2 x^{4/3}}-b n \left (\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\frac {b n \log \left (-e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}\right )}{d}+\frac {\frac {-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e x^{2/3}}{d}\right )}{x^{2/3}}d\left (d+e x^{2/3}\right )-\log \left (1-\frac {d+e x^{2/3}}{d}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{d e x^{2/3}}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 2779 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \left (\frac {\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{2 e^2 x^{4/3}}-b n \left (\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{x^{2/3}}\right )}{x^{2/3}}d\left (d+e x^{2/3}\right )}{d}-\frac {\log \left (1-\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}\right )}{d}+\frac {\frac {-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e x^{2/3}}{d}\right )}{x^{2/3}}d\left (d+e x^{2/3}\right )-\log \left (1-\frac {d+e x^{2/3}}{d}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{d e x^{2/3}}}{d}+\frac {\frac {2 b n \int \frac {\log \left (1-\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{x^{2/3}}d\left (d+e x^{2/3}\right )}{d}-\frac {\log \left (1-\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 2821 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \left (\frac {\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{2 e^2 x^{4/3}}-b n \left (\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{x^{2/3}}\right )}{x^{2/3}}d\left (d+e x^{2/3}\right )}{d}-\frac {\log \left (1-\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}\right )}{d}+\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,\frac {d}{x^{2/3}}\right )}{x^{2/3}}d\left (d+e x^{2/3}\right )\right )}{d}-\frac {\log \left (1-\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{d}}{d}+\frac {-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e x^{2/3}}{d}\right )}{x^{2/3}}d\left (d+e x^{2/3}\right )-\log \left (1-\frac {d+e x^{2/3}}{d}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{d e x^{2/3}}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \left (\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,\frac {d}{x^{2/3}}\right )}{x^{2/3}}d\left (d+e x^{2/3}\right )\right )}{d}-\frac {\log \left (1-\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{d}}{d}+\frac {-\frac {2 b n \left (-\log \left (1-\frac {d+e x^{2/3}}{d}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+e x^{2/3}}{d}\right )\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{d e x^{2/3}}}{d}}{d}+\frac {\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{2 e^2 x^{4/3}}-b n \left (\frac {\frac {b n \log \left (-e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{x^{2/3}}\right )}{d}-\frac {\log \left (1-\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d}}{d}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {3}{2} \left (-b e^3 n \left (\frac {\frac {\left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{2 e^2 x^{4/3}}-b n \left (\frac {\frac {b n \log \left (-e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{x^{2/3}}\right )}{d}-\frac {\log \left (1-\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {2 b n \left (-\log \left (1-\frac {d+e x^{2/3}}{d}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+e x^{2/3}}{d}\right )\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{d e x^{2/3}}}{d}+\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )+b n \operatorname {PolyLog}\left (3,\frac {d}{x^{2/3}}\right )\right )}{d}-\frac {\log \left (1-\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )^2}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{3 x^2}\right )\) |
(3*(-1/3*(a + b*Log[c*(d + e*x^(2/3))^n])^3/x^2 - b*e^3*n*(((a + b*Log[c*x ^((2*n)/3)])^2/(2*e^2*x^(4/3)) - b*n*(((b*n*Log[-(e*x^(2/3))])/d - ((d + e *x^(2/3))*(a + b*Log[c*x^((2*n)/3)]))/(d*e*x^(2/3)))/d + (-((Log[1 - d/x^( 2/3)]*(a + b*Log[c*x^((2*n)/3)]))/d) + (b*n*PolyLog[2, d/x^(2/3)])/d)/d))/ d + ((-(((d + e*x^(2/3))*(a + b*Log[c*x^((2*n)/3)])^2)/(d*e*x^(2/3))) - (2 *b*n*(-(Log[1 - (d + e*x^(2/3))/d]*(a + b*Log[c*x^((2*n)/3)])) - b*n*PolyL og[2, (d + e*x^(2/3))/d]))/d)/d + (-((Log[1 - d/x^(2/3)]*(a + b*Log[c*x^(( 2*n)/3)])^2)/d) + (2*b*n*((a + b*Log[c*x^((2*n)/3)])*PolyLog[2, d/x^(2/3)] + b*n*PolyLog[3, d/x^(2/3)]))/d)/d)/d)))/2
3.5.84.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ .)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{3}}{x^{3}}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
integral((b^3*log((e*x^(2/3) + d)^n*c)^3 + 3*a*b^2*log((e*x^(2/3) + d)^n*c )^2 + 3*a^2*b*log((e*x^(2/3) + d)^n*c) + a^3)/x^3, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
-1/2*b^3*log((e*x^(2/3) + d)^n)^3/x^2 + integrate(((b^3*e*n*x + 3*(b^3*e*l og(c) + a*b^2*e)*x + 3*(b^3*d*log(c) + a*b^2*d)*x^(1/3))*log((e*x^(2/3) + d)^n)^2 + (b^3*e*log(c)^3 + 3*a*b^2*e*log(c)^2 + 3*a^2*b*e*log(c) + a^3*e) *x + 3*((b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x + (b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^(1/3))*log((e*x^(2/3) + d)^n) + (b^3*d*log (c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^2*b*d*log(c) + a^3*d)*x^(1/3))/(e*x^4 + d *x^(10/3)), x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^3}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^3}{x^3} \,d x \]