Integrand size = 23, antiderivative size = 351 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\frac {b^2 n^2}{3 d^3 (d+e x)}+\frac {b^2 n^2 \log (x)}{3 d^4}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {5 b e n x \left (a+b \log \left (c x^n\right )\right )}{3 d^4 (d+e x)}-\frac {5 \left (a+b \log \left (c x^n\right )\right )^2}{6 d^4}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d^2 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )^2}{d^4 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^3}{3 b d^4 n}-\frac {2 b^2 n^2 \log (d+e x)}{d^4}+\frac {11 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^4}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{d^4}+\frac {11 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 d^4}-\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^4}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{d^4} \]
1/3*b^2*n^2/d^3/(e*x+d)+1/3*b^2*n^2*ln(x)/d^4-1/3*b*n*(a+b*ln(c*x^n))/d^2/ (e*x+d)^2+5/3*b*e*n*x*(a+b*ln(c*x^n))/d^4/(e*x+d)-5/6*(a+b*ln(c*x^n))^2/d^ 4+1/3*(a+b*ln(c*x^n))^2/d/(e*x+d)^3+1/2*(a+b*ln(c*x^n))^2/d^2/(e*x+d)^2-e* x*(a+b*ln(c*x^n))^2/d^4/(e*x+d)+1/3*(a+b*ln(c*x^n))^3/b/d^4/n-2*b^2*n^2*ln (e*x+d)/d^4+11/3*b*n*(a+b*ln(c*x^n))*ln(1+e*x/d)/d^4-(a+b*ln(c*x^n))^2*ln( 1+e*x/d)/d^4+11/3*b^2*n^2*polylog(2,-e*x/d)/d^4-2*b*n*(a+b*ln(c*x^n))*poly log(2,-e*x/d)/d^4+2*b^2*n^2*polylog(3,-e*x/d)/d^4
Time = 0.28 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\frac {-\frac {2 b d^2 n \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac {10 b d n \left (a+b \log \left (c x^n\right )\right )}{d+e x}-11 \left (a+b \log \left (c x^n\right )\right )^2+\frac {2 d^3 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}+\frac {3 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}+\frac {6 d \left (a+b \log \left (c x^n\right )\right )^2}{d+e x}+\frac {2 \left (a+b \log \left (c x^n\right )\right )^3}{b n}+10 b^2 n^2 (\log (x)-\log (d+e x))+\frac {2 b^2 n^2 (d+(d+e x) \log (x)-(d+e x) \log (d+e x))}{d+e x}+22 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-6 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )+22 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-12 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )+12 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{6 d^4} \]
((-2*b*d^2*n*(a + b*Log[c*x^n]))/(d + e*x)^2 - (10*b*d*n*(a + b*Log[c*x^n] ))/(d + e*x) - 11*(a + b*Log[c*x^n])^2 + (2*d^3*(a + b*Log[c*x^n])^2)/(d + e*x)^3 + (3*d^2*(a + b*Log[c*x^n])^2)/(d + e*x)^2 + (6*d*(a + b*Log[c*x^n ])^2)/(d + e*x) + (2*(a + b*Log[c*x^n])^3)/(b*n) + 10*b^2*n^2*(Log[x] - Lo g[d + e*x]) + (2*b^2*n^2*(d + (d + e*x)*Log[x] - (d + e*x)*Log[d + e*x]))/ (d + e*x) + 22*b*n*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] - 6*(a + b*Log[c*x^ n])^2*Log[1 + (e*x)/d] + 22*b^2*n^2*PolyLog[2, -((e*x)/d)] - 12*b*n*(a + b *Log[c*x^n])*PolyLog[2, -((e*x)/d)] + 12*b^2*n^2*PolyLog[3, -((e*x)/d)])/( 6*d^4)
Time = 2.79 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.45, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {2789, 2756, 2789, 2756, 54, 2009, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^3}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4}dx}{d}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^3}dx}{d}-\frac {e \left (\frac {2 b n \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3}dx}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3}dx}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3}dx}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \int \frac {1}{x (d+e x)^2}dx}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \int \left (-\frac {e}{d^2 (d+e x)}-\frac {e}{d (d+e x)^2}+\frac {1}{d^2 x}\right )dx}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}dx}{d}}{d}-\frac {e \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2}dx}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2}dx}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}dx}{d}}{d}-\frac {e \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \int \frac {1}{d+e x}dx}{d}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \int \frac {1}{d+e x}dx}{d}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^2}dx}{d}}{d}-\frac {e \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2755 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \int \frac {a+b \log \left (c x^n\right )}{d+e x}dx}{d}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 b n \int \frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (\frac {\frac {b n \int \frac {\log \left (\frac {d}{e x}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\frac {\frac {b n \int \frac {\log \left (\frac {d}{e x}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {b n \int \frac {\log \left (\frac {e x}{d}+1\right )}{x}dx}{e}\right )}{d}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (\frac {\frac {b n \int \frac {\log \left (\frac {d}{e x}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\frac {\frac {b n \int \frac {\log \left (\frac {d}{e x}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \int \frac {\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{x}dx\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}\right )}{d}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {\frac {\frac {\frac {2 b n \left (\operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )+b n \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d (d+e x)}-\frac {2 b n \left (\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e}\right )}{d}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}\right )}{e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {2 b n \left (\frac {\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}\right )}{3 e}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{3 e (d+e x)^3}\right )}{d}\) |
-((e*(-1/3*(a + b*Log[c*x^n])^2/(e*(d + e*x)^3) + (2*b*n*(-((e*(-1/2*(a + b*Log[c*x^n])/(e*(d + e*x)^2) + (b*n*(1/(d*(d + e*x)) + Log[x]/d^2 - Log[d + e*x]/d^2))/(2*e)))/d) + (-((e*((x*(a + b*Log[c*x^n]))/(d*(d + e*x)) - ( b*n*Log[d + e*x])/(d*e)))/d) + (-((Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d) + (b*n*PolyLog[2, -(d/(e*x))])/d)/d)/d))/(3*e)))/d) + (-((e*(-1/2*(a + b* Log[c*x^n])^2/(e*(d + e*x)^2) + (b*n*(-((e*((x*(a + b*Log[c*x^n]))/(d*(d + e*x)) - (b*n*Log[d + e*x])/(d*e)))/d) + (-((Log[1 + d/(e*x)]*(a + b*Log[c *x^n]))/d) + (b*n*PolyLog[2, -(d/(e*x))])/d)/d))/e))/d) + (-((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)/d)/d
3.2.18.3.1 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_) + (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_)), 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[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 = 0.68 (sec) , antiderivative size = 894, normalized size of antiderivative = 2.55
1/4*(-I*b*Pi*csgn(I*c)*csgn(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*(-1/d^4*ln(e*x+d)+1/d^3/(e*x+d)+1/2/d^2/(e*x+d)^2+1/3/d/(e*x+d)^3 +1/d^4*ln(x))-11/3*b^2/d^4*n^2*ln(e*x+d)*ln(-e*x/d)-b^2*n/d^4*ln(x^n)*ln(x )^2-2*b^2/d^4*ln(x)*dilog(-e*x/d)*n^2+2*b^2*n/d^4*ln(x^n)*dilog(-e*x/d)+b^ 2/d^4*n^2*ln(e*x+d)*ln(x)^2-b^2/d^4*n^2*ln(x)^2*ln(1+e*x/d)-2*b^2/d^4*n^2* ln(x)*polylog(2,-e*x/d)+11/3*b^2*n*ln(x^n)/d^4*ln(e*x+d)-11/3*b^2*n*ln(x^n )/d^4*ln(x)+(-I*b*Pi*csgn(I*c)*csgn(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)*b*(-ln(x^n)/d^4*ln(e*x+d)+ln(x^n)/d^3/(e*x+d)+1/2*ln(x^n)/d ^2/(e*x+d)^2+1/3*ln(x^n)/d/(e*x+d)^3+ln(x^n)/d^4*ln(x)-1/6*n*(5/d^3/(e*x+d )+1/d^2/(e*x+d)^2-11/d^4*ln(e*x+d)+11/d^4*ln(x)+3/d^4*ln(x)^2-6/d^4*ln(e*x +d)*ln(-e*x/d)-6/d^4*dilog(-e*x/d)))+2*b^2*n/d^4*ln(x^n)*ln(e*x+d)*ln(-e*x /d)-2*b^2/d^4*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^2-5/3*b^2*n*ln(x^n)/d^3/(e*x+d) -1/3*b^2*n*ln(x^n)/d^2/(e*x+d)^2-b^2*ln(x^n)^2/d^4*ln(e*x+d)+b^2*ln(x^n)^2 /d^3/(e*x+d)+1/2*b^2*ln(x^n)^2/d^2/(e*x+d)^2+1/3*b^2*ln(x^n)^2/d/(e*x+d)^3 +b^2*ln(x^n)^2/d^4*ln(x)+11/6*b^2/d^4*n^2*ln(x)^2-11/3*b^2/d^4*n^2*dilog(- e*x/d)+1/3*b^2/d^4*ln(x)^3*n^2+1/3*b^2*n^2/d^3/(e*x+d)+2*b^2*n^2*ln(x)/d^4 -2*b^2*n^2*ln(e*x+d)/d^4+2*b^2*n^2*polylog(3,-e*x/d)/d^4
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4} x} \,d x } \]
integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^4*x^5 + 4*d*e^3*x^ 4 + 6*d^2*e^2*x^3 + 4*d^3*e*x^2 + d^4*x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{x \left (d + e x\right )^{4}}\, dx \]
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4} x} \,d x } \]
1/6*a^2*((6*e^2*x^2 + 15*d*e*x + 11*d^2)/(d^3*e^3*x^3 + 3*d^4*e^2*x^2 + 3* d^5*e*x + d^6) - 6*log(e*x + d)/d^4 + 6*log(x)/d^4) + integrate((b^2*log(c )^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*log(x^n))/(e^4* x^5 + 4*d*e^3*x^4 + 6*d^2*e^2*x^3 + 4*d^3*e*x^2 + d^4*x), x)
\[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{4} x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)^4} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x\,{\left (d+e\,x\right )}^4} \,d x \]