Integrand size = 23, antiderivative size = 361 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\frac {3 b e n x \left (a+b \log \left (c x^n\right )\right )^2}{2 d^3 (d+e x)}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 d^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 d (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^3}{d^3 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^4}{4 b d^3 n}-\frac {3 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^3}+\frac {9 b n \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{2 d^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {e x}{d}\right )}{d^3}-\frac {3 b^3 n^3 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}+\frac {9 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}-\frac {3 b n \left (a+b \log \left (c x^n\right )\right )^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^3}-\frac {9 b^3 n^3 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^3}+\frac {6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^3}-\frac {6 b^3 n^3 \operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )}{d^3} \]
3/2*b*e*n*x*(a+b*ln(c*x^n))^2/d^3/(e*x+d)-1/2*(a+b*ln(c*x^n))^3/d^3+1/2*(a +b*ln(c*x^n))^3/d/(e*x+d)^2-e*x*(a+b*ln(c*x^n))^3/d^3/(e*x+d)+1/4*(a+b*ln( c*x^n))^4/b/d^3/n-3*b^2*n^2*(a+b*ln(c*x^n))*ln(1+e*x/d)/d^3+9/2*b*n*(a+b*l n(c*x^n))^2*ln(1+e*x/d)/d^3-(a+b*ln(c*x^n))^3*ln(1+e*x/d)/d^3-3*b^3*n^3*po lylog(2,-e*x/d)/d^3+9*b^2*n^2*(a+b*ln(c*x^n))*polylog(2,-e*x/d)/d^3-3*b*n* (a+b*ln(c*x^n))^2*polylog(2,-e*x/d)/d^3-9*b^3*n^3*polylog(3,-e*x/d)/d^3+6* b^2*n^2*(a+b*ln(c*x^n))*polylog(3,-e*x/d)/d^3-6*b^3*n^3*polylog(4,-e*x/d)/ d^3
Time = 0.63 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.96 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\frac {2 d^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3+4 d (d+e x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3+4 (d+e x)^2 \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3-4 (d+e x)^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3 \log (d+e x)+6 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left ((d+e x)^2 \log ^2(x)+(d+e x) (-d+3 (d+e x) \log (d+e x))-\log (x) \left (e x (4 d+3 e x)+2 (d+e x)^2 \log \left (1+\frac {e x}{d}\right )\right )-2 (d+e x)^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )+2 b^2 n^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (-3 e x (2 d+e x) \log ^2(x)+2 (d+e x)^2 \log ^3(x)-6 (d+e x)^2 \log (d+e x)+6 (d+e x) \log (x) \left (e x+(d+e x) \log \left (1+\frac {e x}{d}\right )\right )+6 (d+e x)^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-6 (d+e x) \left (\log (x) \left (e x \log (x)-2 (d+e x) \log \left (1+\frac {e x}{d}\right )\right )-2 (d+e x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )\right )-6 (d+e x)^2 \left (\log ^2(x) \log \left (1+\frac {e x}{d}\right )+2 \log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )\right )+b^3 n^3 \left ((d+e x)^2 \log ^4(x)-4 (d+e x) \left (\log ^2(x) \left (e x \log (x)-3 (d+e x) \log \left (1+\frac {e x}{d}\right )\right )-6 (d+e x) \log (x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 (d+e x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )-2 \left (\log (x) \left (e x (2 d+e x) \log ^2(x)+6 (d+e x)^2 \log \left (1+\frac {e x}{d}\right )-3 (d+e x) \log (x) \left (e x+(d+e x) \log \left (1+\frac {e x}{d}\right )\right )\right )-6 (d+e x)^2 (-1+\log (x)) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+6 (d+e x)^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )-4 (d+e x)^2 \left (\log ^3(x) \log \left (1+\frac {e x}{d}\right )+3 \log ^2(x) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-6 \log (x) \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )+6 \operatorname {PolyLog}\left (4,-\frac {e x}{d}\right )\right )\right )}{4 d^3 (d+e x)^2} \]
(2*d^2*(a - b*n*Log[x] + b*Log[c*x^n])^3 + 4*d*(d + e*x)*(a - b*n*Log[x] + b*Log[c*x^n])^3 + 4*(d + e*x)^2*Log[x]*(a - b*n*Log[x] + b*Log[c*x^n])^3 - 4*(d + e*x)^2*(a - b*n*Log[x] + b*Log[c*x^n])^3*Log[d + e*x] + 6*b*n*(a - b*n*Log[x] + b*Log[c*x^n])^2*((d + e*x)^2*Log[x]^2 + (d + e*x)*(-d + 3*( d + e*x)*Log[d + e*x]) - Log[x]*(e*x*(4*d + 3*e*x) + 2*(d + e*x)^2*Log[1 + (e*x)/d]) - 2*(d + e*x)^2*PolyLog[2, -((e*x)/d)]) + 2*b^2*n^2*(a - b*n*Lo g[x] + b*Log[c*x^n])*(-3*e*x*(2*d + e*x)*Log[x]^2 + 2*(d + e*x)^2*Log[x]^3 - 6*(d + e*x)^2*Log[d + e*x] + 6*(d + e*x)*Log[x]*(e*x + (d + e*x)*Log[1 + (e*x)/d]) + 6*(d + e*x)^2*PolyLog[2, -((e*x)/d)] - 6*(d + e*x)*(Log[x]*( e*x*Log[x] - 2*(d + e*x)*Log[1 + (e*x)/d]) - 2*(d + e*x)*PolyLog[2, -((e*x )/d)]) - 6*(d + e*x)^2*(Log[x]^2*Log[1 + (e*x)/d] + 2*Log[x]*PolyLog[2, -( (e*x)/d)] - 2*PolyLog[3, -((e*x)/d)])) + b^3*n^3*((d + e*x)^2*Log[x]^4 - 4 *(d + e*x)*(Log[x]^2*(e*x*Log[x] - 3*(d + e*x)*Log[1 + (e*x)/d]) - 6*(d + e*x)*Log[x]*PolyLog[2, -((e*x)/d)] + 6*(d + e*x)*PolyLog[3, -((e*x)/d)]) - 2*(Log[x]*(e*x*(2*d + e*x)*Log[x]^2 + 6*(d + e*x)^2*Log[1 + (e*x)/d] - 3* (d + e*x)*Log[x]*(e*x + (d + e*x)*Log[1 + (e*x)/d])) - 6*(d + e*x)^2*(-1 + Log[x])*PolyLog[2, -((e*x)/d)] + 6*(d + e*x)^2*PolyLog[3, -((e*x)/d)]) - 4*(d + e*x)^2*(Log[x]^3*Log[1 + (e*x)/d] + 3*Log[x]^2*PolyLog[2, -((e*x)/d )] - 6*Log[x]*PolyLog[3, -((e*x)/d)] + 6*PolyLog[4, -((e*x)/d)])))/(4*d^3* (d + e*x)^2)
Time = 2.01 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {2789, 2756, 2789, 2755, 2754, 2779, 2821, 2830, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^3}dx}{d}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {3 b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2}dx}{2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 e (d+e x)^2}\right )}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{(d+e x)^2}dx}{d}}{d}-\frac {e \left (\frac {3 b n \left (\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}dx}{d}\right )}{2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 e (d+e x)^2}\right )}{d}\) |
| \(\Big \downarrow \) 2755 |
| \(\displaystyle \frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{d+e x}dx}{d}\right )}{d}}{d}-\frac {e \left (\frac {3 b n \left (\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \int \frac {a+b \log \left (c x^n\right )}{d+e x}dx}{d}\right )}{d}\right )}{2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 e (d+e x)^2}\right )}{d}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}}{d}-\frac {e \left (\frac {3 b n \left (\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}\right )}{2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 e (d+e x)^2}\right )}{d}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {\frac {\frac {3 b n \int \frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}}{d}-\frac {e \left (\frac {3 b n \left (\frac {\frac {2 b n \int \frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}\right )}{2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 e (d+e x)^2}\right )}{d}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {\frac {\frac {3 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \int \frac {\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{e}\right )}{d}\right )}{d}}{d}-\frac {e \left (\frac {3 b n \left (\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}\right )}{2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 e (d+e x)^2}\right )}{d}\) |
| \(\Big \downarrow \) 2830 |
| \(\displaystyle \frac {\frac {\frac {3 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{x}dx-\operatorname {PolyLog}\left (3,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{e}\right )}{d}\right )}{d}}{d}-\frac {e \left (\frac {3 b n \left (\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}\right )}{2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 e (d+e x)^2}\right )}{d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {\frac {3 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{x}dx-\operatorname {PolyLog}\left (3,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \left (b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{x}dx-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{e}\right )}{d}\right )}{d}}{d}-\frac {e \left (\frac {3 b n \left (\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}\right )}{d}\right )}{d}\right )}{2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 e (d+e x)^2}\right )}{d}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\frac {\frac {3 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2-2 b n \left (-\left (\operatorname {PolyLog}\left (3,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )\right )-b n \operatorname {PolyLog}\left (4,-\frac {d}{e x}\right )\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^3}{d (d+e x)}-\frac {3 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 b n \left (b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )-\operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{e}\right )}{d}\right )}{d}}{d}-\frac {e \left (\frac {3 b n \left (\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+b n \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}\right )}{d}\right )}{d}\right )}{2 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^3}{2 e (d+e x)^2}\right )}{d}\) |
-((e*(-1/2*(a + b*Log[c*x^n])^3/(e*(d + e*x)^2) + (3*b*n*(-((e*((x*(a + b* Log[c*x^n])^2)/(d*(d + e*x)) - (2*b*n*(((a + b*Log[c*x^n])*Log[1 + (e*x)/d ])/e + (b*n*PolyLog[2, -((e*x)/d)])/e))/d))/d) + (-((Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^2)/d) + (2*b*n*((a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x))] + b*n*PolyLog[3, -(d/(e*x))]))/d)/d))/(2*e)))/d) + (-((e*((x*(a + b*Log[c*x ^n])^3)/(d*(d + e*x)) - (3*b*n*(((a + b*Log[c*x^n])^2*Log[1 + (e*x)/d])/e - (2*b*n*(-((a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)]) + b*n*PolyLog[3, -( (e*x)/d)]))/e))/d))/d) + (-((Log[1 + d/(e*x)]*(a + b*Log[c*x^n])^3)/d) + ( 3*b*n*((a + b*Log[c*x^n])^2*PolyLog[2, -(d/(e*x))] - 2*b*n*(-((a + b*Log[c *x^n])*PolyLog[3, -(d/(e*x))]) - b*n*PolyLog[4, -(d/(e*x))])))/d)/d)/d
3.2.23.3.1 Defintions of rubi rules used
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[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*PolyLog[k_, (e_.)*(x_)^(q_ .)])/(x_), x_Symbol] :> Simp[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^p/q) , x] - Simp[b*n*(p/q) Int[PolyLog[k + 1, e*x^q]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, e, k, n, q}, x] && GtQ[p, 0]
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[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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.63 (sec) , antiderivative size = 1607, normalized size of antiderivative = 4.45
-6*b^3/d^3*ln(x)*ln(x^n)*ln(e*x+d)*ln(-e*x/d)*n^2+3/4*(-I*b*Pi*csgn(I*c)*c sgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^ n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^2*b*(-ln(x^n)/d^3 *ln(e*x+d)+ln(x^n)/d^2/(e*x+d)+1/2*ln(x^n)/d/(e*x+d)^2+ln(x^n)/d^3*ln(x)-1 /2*n*(1/d^2/(e*x+d)-3/d^3*ln(e*x+d)+3/d^3*ln(x)+1/d^3*ln(x)^2-2/d^3*ln(e*x +d)*ln(-e*x/d)-2/d^3*dilog(-e*x/d)))+3/2*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*cs gn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*csgn(I*c*x ^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b^2*(-ln(x^n)^2/d^3*ln(e*x+d)+ ln(x^n)^2/d^2/(e*x+d)+1/2*ln(x^n)^2/d/(e*x+d)^2+ln(x^n)^2/d^3*ln(x)-n*(1/d ^2*(ln(x^n)/(e*x+d)-3*ln(x^n)/d*ln(e*x+d)+3*ln(x^n)/d*ln(x)-n*(-1/d*ln(e*x +d)+1/d*ln(x)+3/2/d*ln(x)^2-3/d*ln(e*x+d)*ln(-e*x/d)-3/d*dilog(-e*x/d)))+1 /d^3*ln(x^n)*ln(x)^2-1/3/d^3*ln(x)^3*n-2/d^3*((ln(x^n)-n*ln(x))*(dilog(-e* x/d)+ln(e*x+d)*ln(-e*x/d))+n*(1/2*ln(e*x+d)*ln(x)^2-1/2*ln(x)^2*ln(1+e*x/d )-ln(x)*polylog(2,-e*x/d)+polylog(3,-e*x/d)))))+1/8*(-I*b*Pi*csgn(I*c)*csg n(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n) *csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^3*(-1/d^3*ln(e*x+d) +1/d^2/(e*x+d)+1/2/d/(e*x+d)^2+1/d^3*ln(x))-9/2*b^3/d^3*n^3*ln(e*x+d)*ln(x )^2-3/2*b^3/d^3*n^3*ln(x)^2+3*b^3/d^3*n^3*dilog(-e*x/d)-3/2*b^3/d^3*ln(x)^ 3*n^3-1/4*b^3/d^3*ln(x)^4*n^3-b^3*ln(x^n)^3/d^3*ln(e*x+d)+b^3*ln(x^n)^3/d^ 2/(e*x+d)+1/2*b^3*ln(x^n)^3/d/(e*x+d)^2+b^3*ln(x^n)^3/d^3*ln(x)+9*b^3/d...
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3} x} \,d x } \]
integral((b^3*log(c*x^n)^3 + 3*a*b^2*log(c*x^n)^2 + 3*a^2*b*log(c*x^n) + a ^3)/(e^3*x^4 + 3*d*e^2*x^3 + 3*d^2*e*x^2 + d^3*x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{3}}{x \left (d + e x\right )^{3}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3} x} \,d x } \]
1/2*a^3*((2*e*x + 3*d)/(d^2*e^2*x^2 + 2*d^3*e*x + d^4) - 2*log(e*x + d)/d^ 3 + 2*log(x)/d^3) + integrate((b^3*log(c)^3 + b^3*log(x^n)^3 + 3*a*b^2*log (c)^2 + 3*a^2*b*log(c) + 3*(b^3*log(c) + a*b^2)*log(x^n)^2 + 3*(b^3*log(c) ^2 + 2*a*b^2*log(c) + a^2*b)*log(x^n))/(e^3*x^4 + 3*d*e^2*x^3 + 3*d^2*e*x^ 2 + d^3*x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{x (d+e x)^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x\,{\left (d+e\,x\right )}^3} \,d x \]