Integrand size = 24, antiderivative size = 412 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=-\frac {b^2 e^2 n^2}{40 d^2 x^{8/3}}+\frac {3 b^2 e^3 n^2}{40 d^3 x^2}-\frac {47 b^2 e^4 n^2}{240 d^4 x^{4/3}}+\frac {77 b^2 e^5 n^2}{120 d^5 x^{2/3}}-\frac {77 b^2 e^6 n^2 \log \left (d+e x^{2/3}\right )}{120 d^6}-\frac {b e n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{10 d x^{10/3}}+\frac {b e^2 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{8 d^2 x^{8/3}}-\frac {b e^3 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{6 d^3 x^2}+\frac {b e^4 n \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{4 d^4 x^{4/3}}-\frac {b e^5 n \left (d+e x^{2/3}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6 x^{2/3}}-\frac {b e^6 n \log \left (1-\frac {d}{d+e x^{2/3}}\right ) \left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )}{2 d^6}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}+\frac {137 b^2 e^6 n^2 \log (x)}{180 d^6}+\frac {b^2 e^6 n^2 \operatorname {PolyLog}\left (2,\frac {d}{d+e x^{2/3}}\right )}{2 d^6} \]
-1/40*b^2*e^2*n^2/d^2/x^(8/3)+3/40*b^2*e^3*n^2/d^3/x^2-47/240*b^2*e^4*n^2/ d^4/x^(4/3)+77/120*b^2*e^5*n^2/d^5/x^(2/3)-77/120*b^2*e^6*n^2*ln(d+e*x^(2/ 3))/d^6-1/10*b*e*n*(a+b*ln(c*(d+e*x^(2/3))^n))/d/x^(10/3)+1/8*b*e^2*n*(a+b *ln(c*(d+e*x^(2/3))^n))/d^2/x^(8/3)-1/6*b*e^3*n*(a+b*ln(c*(d+e*x^(2/3))^n) )/d^3/x^2+1/4*b*e^4*n*(a+b*ln(c*(d+e*x^(2/3))^n))/d^4/x^(4/3)-1/2*b*e^5*n* (d+e*x^(2/3))*(a+b*ln(c*(d+e*x^(2/3))^n))/d^6/x^(2/3)-1/2*b*e^6*n*ln(1-d/( d+e*x^(2/3)))*(a+b*ln(c*(d+e*x^(2/3))^n))/d^6-1/4*(a+b*ln(c*(d+e*x^(2/3))^ n))^2/x^4+137/180*b^2*e^6*n^2*ln(x)/d^6+1/2*b^2*e^6*n^2*polylog(2,d/(d+e*x ^(2/3)))/d^6
Time = 0.33 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{4 x^4}-\frac {b e \left (72 a d^5 n-90 a d^4 e n x^{2/3}+18 b d^4 e n^2 x^{2/3}+120 a d^3 e^2 n x^{4/3}-54 b d^3 e^2 n^2 x^{4/3}-180 a d^2 e^3 n x^2+141 b d^2 e^3 n^2 x^2+360 a d e^4 n x^{8/3}-462 b d e^4 n^2 x^{8/3}+6 e^5 n (-60 a+137 b n) x^{10/3} \log \left (d+e x^{2/3}\right )+72 b d^5 n \log \left (c \left (d+e x^{2/3}\right )^n\right )-90 b d^4 e n x^{2/3} \log \left (c \left (d+e x^{2/3}\right )^n\right )+120 b d^3 e^2 n x^{4/3} \log \left (c \left (d+e x^{2/3}\right )^n\right )-180 b d^2 e^3 n x^2 \log \left (c \left (d+e x^{2/3}\right )^n\right )+360 b d e^4 n x^{8/3} \log \left (c \left (d+e x^{2/3}\right )^n\right )-180 b e^5 x^{10/3} \log ^2\left (c \left (d+e x^{2/3}\right )^n\right )+360 b e^5 n x^{10/3} \log \left (c \left (d+e x^{2/3}\right )^n\right ) \log \left (-\frac {e x^{2/3}}{d}\right )+240 a e^5 n x^{10/3} \log (x)-548 b e^5 n^2 x^{10/3} \log (x)+360 b e^5 n^2 x^{10/3} \operatorname {PolyLog}\left (2,1+\frac {e x^{2/3}}{d}\right )\right )}{720 d^6 x^{10/3}} \]
-1/4*(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^4 - (b*e*(72*a*d^5*n - 90*a*d^4* e*n*x^(2/3) + 18*b*d^4*e*n^2*x^(2/3) + 120*a*d^3*e^2*n*x^(4/3) - 54*b*d^3* e^2*n^2*x^(4/3) - 180*a*d^2*e^3*n*x^2 + 141*b*d^2*e^3*n^2*x^2 + 360*a*d*e^ 4*n*x^(8/3) - 462*b*d*e^4*n^2*x^(8/3) + 6*e^5*n*(-60*a + 137*b*n)*x^(10/3) *Log[d + e*x^(2/3)] + 72*b*d^5*n*Log[c*(d + e*x^(2/3))^n] - 90*b*d^4*e*n*x ^(2/3)*Log[c*(d + e*x^(2/3))^n] + 120*b*d^3*e^2*n*x^(4/3)*Log[c*(d + e*x^( 2/3))^n] - 180*b*d^2*e^3*n*x^2*Log[c*(d + e*x^(2/3))^n] + 360*b*d*e^4*n*x^ (8/3)*Log[c*(d + e*x^(2/3))^n] - 180*b*e^5*x^(10/3)*Log[c*(d + e*x^(2/3))^ n]^2 + 360*b*e^5*n*x^(10/3)*Log[c*(d + e*x^(2/3))^n]*Log[-((e*x^(2/3))/d)] + 240*a*e^5*n*x^(10/3)*Log[x] - 548*b*e^5*n^2*x^(10/3)*Log[x] + 360*b*e^5 *n^2*x^(10/3)*PolyLog[2, 1 + (e*x^(2/3))/d]))/(720*d^6*x^(10/3))
Time = 2.21 (sec) , antiderivative size = 568, normalized size of antiderivative = 1.38, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.042, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle \frac {3}{2} \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^{14/3}}dx^{2/3}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e n \int \frac {a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )}{\left (d+e x^{2/3}\right ) x^4}dx^{2/3}-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b n \int \frac {a+b \log \left (c x^{2 n/3}\right )}{x^{14/3}}d\left (d+e x^{2/3}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^6 x^{14/3}}d\left (d+e x^{2/3}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^6 x^4}d\left (d+e x^{2/3}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e^5 x^4}d\left (d+e x^{2/3}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {-\frac {1}{5} b n \int -\frac {1}{e^5 x^4}d\left (d+e x^{2/3}\right )-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e^5 x^4}d\left (d+e x^{2/3}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {-\frac {1}{5} b n \int \left (-\frac {1}{d^5 e x^{2/3}}+\frac {1}{d^5 x^{2/3}}+\frac {1}{d^4 e^2 x^{4/3}}-\frac {1}{d^3 e^3 x^2}+\frac {1}{d^2 e^4 x^{8/3}}-\frac {1}{d e^5 x^{10/3}}\right )d\left (d+e x^{2/3}\right )-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e^5 x^4}d\left (d+e x^{2/3}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e^5 x^4}d\left (d+e x^{2/3}\right )}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e^5 x^{10/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^4 x^{10/3}}d\left (d+e x^{2/3}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \int \frac {1}{e^4 x^{10/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^4 x^{10/3}}d\left (d+e x^{2/3}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \int \left (-\frac {1}{d^4 e x^{2/3}}+\frac {1}{d^4 x^{2/3}}+\frac {1}{d^3 e^2 x^{4/3}}-\frac {1}{d^2 e^3 x^2}+\frac {1}{d e^4 x^{8/3}}\right )d\left (d+e x^{2/3}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^4 x^{10/3}}d\left (d+e x^{2/3}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^4 x^{10/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^4 x^{8/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e^3 x^{8/3}}d\left (d+e x^{2/3}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e^3 x^{8/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {-\frac {1}{3} b n \int -\frac {1}{e^3 x^{8/3}}d\left (d+e x^{2/3}\right )-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {-\frac {1}{3} b n \int \left (-\frac {1}{d^3 e x^{2/3}}+\frac {1}{d^3 x^{2/3}}+\frac {1}{d^2 e^2 x^{4/3}}-\frac {1}{d e^3 x^2}\right )d\left (d+e x^{2/3}\right )-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}}{d}+\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e^3 x^{8/3}}d\left (d+e x^{2/3}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e^3 x^{8/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}-\frac {1}{3} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^3}-\frac {\log \left (-e x^{2/3}\right )}{d^3}-\frac {1}{d^2 e x^{2/3}}+\frac {1}{2 d e^2 x^{4/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e^3 x^2}d\left (d+e x^{2/3}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^2 x^2}d\left (d+e x^{2/3}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}-\frac {1}{3} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^3}-\frac {\log \left (-e x^{2/3}\right )}{d^3}-\frac {1}{d^2 e x^{2/3}}+\frac {1}{2 d e^2 x^{4/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{2 e^2 x^{4/3}}-\frac {1}{2} b n \int \frac {1}{e^2 x^2}d\left (d+e x^{2/3}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^2 x^2}d\left (d+e x^{2/3}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}-\frac {1}{3} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^3}-\frac {\log \left (-e x^{2/3}\right )}{d^3}-\frac {1}{d^2 e x^{2/3}}+\frac {1}{2 d e^2 x^{4/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{2 e^2 x^{4/3}}-\frac {1}{2} b n \int \left (-\frac {1}{d^2 e x^{2/3}}+\frac {1}{d^2 x^{2/3}}+\frac {1}{d e^2 x^{4/3}}\right )d\left (d+e x^{2/3}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^2 x^2}d\left (d+e x^{2/3}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}-\frac {1}{3} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^3}-\frac {\log \left (-e x^{2/3}\right )}{d^3}-\frac {1}{d^2 e x^{2/3}}+\frac {1}{2 d e^2 x^{4/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^2 x^2}d\left (d+e x^{2/3}\right )}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{2 e^2 x^{4/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^2}-\frac {\log \left (-e x^{2/3}\right )}{d^2}-\frac {1}{d e x^{2/3}}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}-\frac {1}{3} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^3}-\frac {\log \left (-e x^{2/3}\right )}{d^3}-\frac {1}{d^2 e x^{2/3}}+\frac {1}{2 d e^2 x^{4/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^{2 n/3}\right )}{e^2 x^{4/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{2 e^2 x^{4/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^2}-\frac {\log \left (-e x^{2/3}\right )}{d^2}-\frac {1}{d e x^{2/3}}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}-\frac {1}{3} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^3}-\frac {\log \left (-e x^{2/3}\right )}{d^3}-\frac {1}{d^2 e x^{2/3}}+\frac {1}{2 d e^2 x^{4/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {-\frac {b n \int -\frac {1}{e x^{2/3}}d\left (d+e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{2 e^2 x^{4/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^2}-\frac {\log \left (-e x^{2/3}\right )}{d^2}-\frac {1}{d e x^{2/3}}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}-\frac {1}{3} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^3}-\frac {\log \left (-e x^{2/3}\right )}{d^3}-\frac {1}{d^2 e x^{2/3}}+\frac {1}{2 d e^2 x^{4/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {\int -\frac {a+b \log \left (c x^{2 n/3}\right )}{e x^{4/3}}d\left (d+e x^{2/3}\right )}{d}+\frac {\frac {b n \log \left (-e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{2 e^2 x^{4/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^2}-\frac {\log \left (-e x^{2/3}\right )}{d^2}-\frac {1}{d e x^{2/3}}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}-\frac {1}{3} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^3}-\frac {\log \left (-e x^{2/3}\right )}{d^3}-\frac {1}{d^2 e x^{2/3}}+\frac {1}{2 d e^2 x^{4/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{x^{2/3}}\right )}{x^{2/3}}d\left (d+e x^{2/3}\right )}{d}-\frac {\log \left (1-\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d}}{d}+\frac {\frac {b n \log \left (-e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{2 e^2 x^{4/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^2}-\frac {\log \left (-e x^{2/3}\right )}{d^2}-\frac {1}{d e x^{2/3}}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}-\frac {1}{3} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^3}-\frac {\log \left (-e x^{2/3}\right )}{d^3}-\frac {1}{d^2 e x^{2/3}}+\frac {1}{2 d e^2 x^{4/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {3}{2} \left (\frac {1}{3} b e^6 n \left (\frac {\frac {\frac {\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{2 e^2 x^{4/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^2}-\frac {\log \left (-e x^{2/3}\right )}{d^2}-\frac {1}{d e x^{2/3}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e x^{2/3}\right )}{d}-\frac {\left (d+e x^{2/3}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d e x^{2/3}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{x^{2/3}}\right )}{d}-\frac {\log \left (1-\frac {d}{x^{2/3}}\right ) \left (a+b \log \left (c x^{2 n/3}\right )\right )}{d}}{d}}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{3 e^3 x^2}-\frac {1}{3} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^3}-\frac {\log \left (-e x^{2/3}\right )}{d^3}-\frac {1}{d^2 e x^{2/3}}+\frac {1}{2 d e^2 x^{4/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{2 n/3}\right )}{4 e^4 x^{8/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^4}-\frac {\log \left (-e x^{2/3}\right )}{d^4}-\frac {1}{d^3 e x^{2/3}}+\frac {1}{2 d^2 e^2 x^{4/3}}-\frac {1}{3 d e^3 x^2}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{2 n/3}\right )}{5 e^5 x^{10/3}}-\frac {1}{5} b n \left (\frac {\log \left (d+e x^{2/3}\right )}{d^5}-\frac {\log \left (-e x^{2/3}\right )}{d^5}-\frac {1}{d^4 e x^{2/3}}+\frac {1}{2 d^3 e^2 x^{4/3}}-\frac {1}{3 d^2 e^3 x^2}+\frac {1}{4 d e^4 x^{8/3}}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{6 x^4}\right )\) |
(3*(-1/6*(a + b*Log[c*(d + e*x^(2/3))^n])^2/x^4 + (b*e^6*n*((-1/5*(b*n*(1/ (4*d*e^4*x^(8/3)) - 1/(3*d^2*e^3*x^2) + 1/(2*d^3*e^2*x^(4/3)) - 1/(d^4*e*x ^(2/3)) + Log[d + e*x^(2/3)]/d^5 - Log[-(e*x^(2/3))]/d^5)) - (a + b*Log[c* x^((2*n)/3)])/(5*e^5*x^(10/3)))/d + ((-1/4*(b*n*(-1/3*1/(d*e^3*x^2) + 1/(2 *d^2*e^2*x^(4/3)) - 1/(d^3*e*x^(2/3)) + Log[d + e*x^(2/3)]/d^4 - Log[-(e*x ^(2/3))]/d^4)) + (a + b*Log[c*x^((2*n)/3)])/(4*e^4*x^(8/3)))/d + ((-1/3*(b *n*(1/(2*d*e^2*x^(4/3)) - 1/(d^2*e*x^(2/3)) + Log[d + e*x^(2/3)]/d^3 - Log [-(e*x^(2/3))]/d^3)) - (a + b*Log[c*x^((2*n)/3)])/(3*e^3*x^2))/d + ((-1/2* (b*n*(-(1/(d*e*x^(2/3))) + Log[d + e*x^(2/3)]/d^2 - Log[-(e*x^(2/3))]/d^2) ) + (a + b*Log[c*x^((2*n)/3)])/(2*e^2*x^(4/3)))/d + (((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)/d)/d))/3))/2
3.5.75.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(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])
\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {2}{3}}\right )^{n}\right )\right )}^{2}}{x^{5}}d x\]
\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{5}} \,d x } \]
-1/4*b^2*log((e*x^(2/3) + d)^n)^2/x^4 + integrate(1/3*(3*(b^2*e*log(c)^2 + 2*a*b*e*log(c) + a^2*e)*x + (b^2*e*n*x + 6*(b^2*e*log(c) + a*b*e)*x + 6*( b^2*d*log(c) + a*b*d)*x^(1/3))*log((e*x^(2/3) + d)^n) + 3*(b^2*d*log(c)^2 + 2*a*b*d*log(c) + a^2*d)*x^(1/3))/(e*x^6 + d*x^(16/3)), x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {2}{3}} + d\right )}^{n} c\right ) + a\right )}^{2}}{x^{5}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e x^{2/3}\right )^n\right )\right )^2}{x^5} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{2/3}\right )}^n\right )\right )}^2}{x^5} \,d x \]