Integrand size = 21, antiderivative size = 294 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {29 b n}{20 d^6 (d+e x)}-\frac {29 b n \log (x)}{20 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^7}+\frac {49 b n \log (d+e x)}{20 d^7}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^7} \]
-1/30*b*n/d^2/(e*x+d)^5-11/120*b*n/d^3/(e*x+d)^4-37/180*b*n/d^4/(e*x+d)^3- 19/40*b*n/d^5/(e*x+d)^2-29/20*b*n/d^6/(e*x+d)-29/20*b*n*ln(x)/d^7+1/6*(a+b *ln(c*x^n))/d/(e*x+d)^6+1/5*(a+b*ln(c*x^n))/d^2/(e*x+d)^5+1/4*(a+b*ln(c*x^ n))/d^3/(e*x+d)^4+1/3*(a+b*ln(c*x^n))/d^4/(e*x+d)^3+1/2*(a+b*ln(c*x^n))/d^ 5/(e*x+d)^2-e*x*(a+b*ln(c*x^n))/d^7/(e*x+d)-ln(1+d/e/x)*(a+b*ln(c*x^n))/d^ 7+49/20*b*n*ln(e*x+d)/d^7+b*n*polylog(2,-d/e/x)/d^7
Time = 0.25 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\frac {\frac {60 a d^6}{(d+e x)^6}+\frac {72 a d^5}{(d+e x)^5}-\frac {12 b d^5 n}{(d+e x)^5}+\frac {90 a d^4}{(d+e x)^4}-\frac {33 b d^4 n}{(d+e x)^4}+\frac {120 a d^3}{(d+e x)^3}-\frac {74 b d^3 n}{(d+e x)^3}+\frac {180 a d^2}{(d+e x)^2}-\frac {171 b d^2 n}{(d+e x)^2}+\frac {360 a d}{d+e x}-\frac {522 b d n}{d+e x}-882 b n \log (x)+\frac {360 a \log \left (c x^n\right )}{n}+\frac {60 b d^6 \log \left (c x^n\right )}{(d+e x)^6}+\frac {72 b d^5 \log \left (c x^n\right )}{(d+e x)^5}+\frac {90 b d^4 \log \left (c x^n\right )}{(d+e x)^4}+\frac {120 b d^3 \log \left (c x^n\right )}{(d+e x)^3}+\frac {180 b d^2 \log \left (c x^n\right )}{(d+e x)^2}+\frac {360 b d \log \left (c x^n\right )}{d+e x}+\frac {180 b \log ^2\left (c x^n\right )}{n}+882 b n \log (d+e x)-360 a \log \left (1+\frac {e x}{d}\right )-360 b \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )-360 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{360 d^7} \]
((60*a*d^6)/(d + e*x)^6 + (72*a*d^5)/(d + e*x)^5 - (12*b*d^5*n)/(d + e*x)^ 5 + (90*a*d^4)/(d + e*x)^4 - (33*b*d^4*n)/(d + e*x)^4 + (120*a*d^3)/(d + e *x)^3 - (74*b*d^3*n)/(d + e*x)^3 + (180*a*d^2)/(d + e*x)^2 - (171*b*d^2*n) /(d + e*x)^2 + (360*a*d)/(d + e*x) - (522*b*d*n)/(d + e*x) - 882*b*n*Log[x ] + (360*a*Log[c*x^n])/n + (60*b*d^6*Log[c*x^n])/(d + e*x)^6 + (72*b*d^5*L og[c*x^n])/(d + e*x)^5 + (90*b*d^4*Log[c*x^n])/(d + e*x)^4 + (120*b*d^3*Lo g[c*x^n])/(d + e*x)^3 + (180*b*d^2*Log[c*x^n])/(d + e*x)^2 + (360*b*d*Log[ c*x^n])/(d + e*x) + (180*b*Log[c*x^n]^2)/n + 882*b*n*Log[d + e*x] - 360*a* Log[1 + (e*x)/d] - 360*b*Log[c*x^n]*Log[1 + (e*x)/d] - 360*b*n*PolyLog[2, -((e*x)/d)])/(360*d^7)
Leaf count is larger than twice the leaf count of optimal. \(604\) vs. \(2(294)=588\).
Time = 2.64 (sec) , antiderivative size = 604, normalized size of antiderivative = 2.05, number of steps used = 25, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.190, Rules used = {2789, 2756, 54, 2009, 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.
| \(\displaystyle \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^6}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7}dx}{d}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^6}dx}{d}-\frac {e \left (\frac {b n \int \frac {1}{x (d+e x)^6}dx}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^6}dx}{d}-\frac {e \left (\frac {b n \int \left (-\frac {e}{d^6 (d+e x)}-\frac {e}{d^5 (d+e x)^2}-\frac {e}{d^4 (d+e x)^3}-\frac {e}{d^3 (d+e x)^4}-\frac {e}{d^2 (d+e x)^5}-\frac {e}{d (d+e x)^6}+\frac {1}{d^6 x}\right )dx}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^6}dx}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^5}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^6}dx}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^5}dx}{d}-\frac {e \left (\frac {b n \int \frac {1}{x (d+e x)^5}dx}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^5}dx}{d}-\frac {e \left (\frac {b n \int \left (-\frac {e}{d^5 (d+e x)}-\frac {e}{d^4 (d+e x)^2}-\frac {e}{d^3 (d+e x)^3}-\frac {e}{d^2 (d+e x)^4}-\frac {e}{d (d+e x)^5}+\frac {1}{d^5 x}\right )dx}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^5}dx}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^5}dx}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4}dx}{d}-\frac {e \left (\frac {b n \int \frac {1}{x (d+e x)^4}dx}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4}dx}{d}-\frac {e \left (\frac {b n \int \left (-\frac {e}{d^4 (d+e x)}-\frac {e}{d^3 (d+e x)^2}-\frac {e}{d^2 (d+e x)^3}-\frac {e}{d (d+e x)^4}+\frac {1}{d^4 x}\right )dx}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4}dx}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4}dx}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3}dx}{d}-\frac {e \left (\frac {b n \int \frac {1}{x (d+e x)^3}dx}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3}dx}{d}-\frac {e \left (\frac {b n \int \left (-\frac {e}{d^3 (d+e x)}-\frac {e}{d^2 (d+e x)^2}-\frac {e}{d (d+e x)^3}+\frac {1}{d^3 x}\right )dx}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3}dx}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3}dx}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \int \frac {1}{x (d+e x)^2}dx}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \int \left (-\frac {e}{d^2 (d+e x)}-\frac {e}{d (d+e x)^2}+\frac {1}{d^2 x}\right )dx}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2}dx}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2}dx}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \int \frac {1}{d+e x}dx}{d}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)}dx}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {b n \int \frac {\log \left (\frac {d}{e x}+1\right )}{x}dx}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {\frac {\frac {\frac {\frac {\frac {\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d}-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d}}{d}-\frac {e \left (\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^2}+\frac {\log (x)}{d^2}+\frac {1}{d (d+e x)}\right )}{2 e}-\frac {a+b \log \left (c x^n\right )}{2 e (d+e x)^2}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^3}+\frac {\log (x)}{d^3}+\frac {1}{d^2 (d+e x)}+\frac {1}{2 d (d+e x)^2}\right )}{3 e}-\frac {a+b \log \left (c x^n\right )}{3 e (d+e x)^3}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^4}+\frac {\log (x)}{d^4}+\frac {1}{d^3 (d+e x)}+\frac {1}{2 d^2 (d+e x)^2}+\frac {1}{3 d (d+e x)^3}\right )}{4 e}-\frac {a+b \log \left (c x^n\right )}{4 e (d+e x)^4}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^5}+\frac {\log (x)}{d^5}+\frac {1}{d^4 (d+e x)}+\frac {1}{2 d^3 (d+e x)^2}+\frac {1}{3 d^2 (d+e x)^3}+\frac {1}{4 d (d+e x)^4}\right )}{5 e}-\frac {a+b \log \left (c x^n\right )}{5 e (d+e x)^5}\right )}{d}}{d}-\frac {e \left (\frac {b n \left (-\frac {\log (d+e x)}{d^6}+\frac {\log (x)}{d^6}+\frac {1}{d^5 (d+e x)}+\frac {1}{2 d^4 (d+e x)^2}+\frac {1}{3 d^3 (d+e x)^3}+\frac {1}{4 d^2 (d+e x)^4}+\frac {1}{5 d (d+e x)^5}\right )}{6 e}-\frac {a+b \log \left (c x^n\right )}{6 e (d+e x)^6}\right )}{d}\) |
-((e*(-1/6*(a + b*Log[c*x^n])/(e*(d + e*x)^6) + (b*n*(1/(5*d*(d + e*x)^5) + 1/(4*d^2*(d + e*x)^4) + 1/(3*d^3*(d + e*x)^3) + 1/(2*d^4*(d + e*x)^2) + 1/(d^5*(d + e*x)) + Log[x]/d^6 - Log[d + e*x]/d^6))/(6*e)))/d) + (-((e*(-1 /5*(a + b*Log[c*x^n])/(e*(d + e*x)^5) + (b*n*(1/(4*d*(d + e*x)^4) + 1/(3*d ^2*(d + e*x)^3) + 1/(2*d^3*(d + e*x)^2) + 1/(d^4*(d + e*x)) + Log[x]/d^5 - Log[d + e*x]/d^5))/(5*e)))/d) + (-((e*(-1/4*(a + b*Log[c*x^n])/(e*(d + e* x)^4) + (b*n*(1/(3*d*(d + e*x)^3) + 1/(2*d^2*(d + e*x)^2) + 1/(d^3*(d + e* x)) + Log[x]/d^4 - Log[d + e*x]/d^4))/(4*e)))/d) + (-((e*(-1/3*(a + b*Log[ c*x^n])/(e*(d + e*x)^3) + (b*n*(1/(2*d*(d + e*x)^2) + 1/(d^2*(d + e*x)) + Log[x]/d^3 - Log[d + e*x]/d^3))/(3*e)))/d) + (-((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*Poly Log[2, -(d/(e*x))])/d)/d)/d)/d)/d)/d)/d
3.1.71.3.1 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_))^(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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.21 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.51
| method | result | size |
| risch | \(-\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{7}}+\frac {b \ln \left (x^{n}\right )}{d^{6} \left (e x +d \right )}+\frac {b \ln \left (x^{n}\right )}{2 d^{5} \left (e x +d \right )^{2}}+\frac {b \ln \left (x^{n}\right )}{3 d^{4} \left (e x +d \right )^{3}}+\frac {b \ln \left (x^{n}\right )}{4 d^{3} \left (e x +d \right )^{4}}+\frac {b \ln \left (x^{n}\right )}{5 d^{2} \left (e x +d \right )^{5}}+\frac {b \ln \left (x^{n}\right )}{6 d \left (e x +d \right )^{6}}+\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{7}}-\frac {29 b n}{20 d^{6} \left (e x +d \right )}-\frac {19 b n}{40 d^{5} \left (e x +d \right )^{2}}-\frac {37 b n}{180 d^{4} \left (e x +d \right )^{3}}-\frac {11 b n}{120 d^{3} \left (e x +d \right )^{4}}-\frac {b n}{30 d^{2} \left (e x +d \right )^{5}}+\frac {49 b n \ln \left (e x +d \right )}{20 d^{7}}-\frac {49 b n \ln \left (x \right )}{20 d^{7}}-\frac {b n \ln \left (x \right )^{2}}{2 d^{7}}+\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{7}}+\frac {b n \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{7}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (e x +d \right )}{d^{7}}+\frac {1}{d^{6} \left (e x +d \right )}+\frac {1}{2 d^{5} \left (e x +d \right )^{2}}+\frac {1}{3 d^{4} \left (e x +d \right )^{3}}+\frac {1}{4 d^{3} \left (e x +d \right )^{4}}+\frac {1}{5 d^{2} \left (e x +d \right )^{5}}+\frac {1}{6 d \left (e x +d \right )^{6}}+\frac {\ln \left (x \right )}{d^{7}}\right )\) | \(445\) |
-b*ln(x^n)/d^7*ln(e*x+d)+b*ln(x^n)/d^6/(e*x+d)+1/2*b*ln(x^n)/d^5/(e*x+d)^2 +1/3*b*ln(x^n)/d^4/(e*x+d)^3+1/4*b*ln(x^n)/d^3/(e*x+d)^4+1/5*b*ln(x^n)/d^2 /(e*x+d)^5+1/6*b*ln(x^n)/d/(e*x+d)^6+b*ln(x^n)/d^7*ln(x)-29/20*b*n/d^6/(e* x+d)-19/40*b*n/d^5/(e*x+d)^2-37/180*b*n/d^4/(e*x+d)^3-11/120*b*n/d^3/(e*x+ d)^4-1/30*b*n/d^2/(e*x+d)^5+49/20*b*n*ln(e*x+d)/d^7-49/20*b*n*ln(x)/d^7-1/ 2*b*n/d^7*ln(x)^2+b*n/d^7*ln(e*x+d)*ln(-e*x/d)+b*n/d^7*dilog(-e*x/d)+(-1/2 *I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c* x^n)^2+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*I*b*Pi*csgn(I*c*x^n)^3+b *ln(c)+a)*(-1/d^7*ln(e*x+d)+1/d^6/(e*x+d)+1/2/d^5/(e*x+d)^2+1/3/d^4/(e*x+d )^3+1/4/d^3/(e*x+d)^4+1/5/d^2/(e*x+d)^5+1/6/d/(e*x+d)^6+1/d^7*ln(x))
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x} \,d x } \]
integral((b*log(c*x^n) + a)/(e^7*x^8 + 7*d*e^6*x^7 + 21*d^2*e^5*x^6 + 35*d ^3*e^4*x^5 + 35*d^4*e^3*x^4 + 21*d^5*e^2*x^3 + 7*d^6*e*x^2 + d^7*x), x)
Time = 156.44 (sec) , antiderivative size = 1518, normalized size of antiderivative = 5.16 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\text {Too large to display} \]
-a*e*Piecewise((x/d**7, Eq(e, 0)), (-1/(6*e*(d + e*x)**6), True))/d - a*e* Piecewise((x/d**6, Eq(e, 0)), (-1/(5*e*(d + e*x)**5), True))/d**2 - a*e*Pi ecewise((x/d**5, Eq(e, 0)), (-1/(4*e*(d + e*x)**4), True))/d**3 - a*e*Piec ewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/d**4 - a*e*Piecew ise((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))/d**5 - a*e*Piecewis e((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/d**6 - a*e*Piecewise((x/d , Eq(e, 0)), (log(d + e*x)/e, True))/d**7 + a*log(x)/d**7 + b*e**6*n*Piece wise((-1/(e**7*x), Eq(d, 0)), (-137*d**4/(360*d**5*e**6 + 1800*d**4*e**7*x + 3600*d**3*e**8*x**2 + 3600*d**2*e**9*x**3 + 1800*d*e**10*x**4 + 360*e** 11*x**5) - 625*d**3*e*x/(360*d**5*e**6 + 1800*d**4*e**7*x + 3600*d**3*e**8 *x**2 + 3600*d**2*e**9*x**3 + 1800*d*e**10*x**4 + 360*e**11*x**5) - 1100*d **2*e**2*x**2/(360*d**5*e**6 + 1800*d**4*e**7*x + 3600*d**3*e**8*x**2 + 36 00*d**2*e**9*x**3 + 1800*d*e**10*x**4 + 360*e**11*x**5) - 900*d*e**3*x**3/ (360*d**5*e**6 + 1800*d**4*e**7*x + 3600*d**3*e**8*x**2 + 3600*d**2*e**9*x **3 + 1800*d*e**10*x**4 + 360*e**11*x**5) - 300*e**4*x**4/(360*d**5*e**6 + 1800*d**4*e**7*x + 3600*d**3*e**8*x**2 + 3600*d**2*e**9*x**3 + 1800*d*e** 10*x**4 + 360*e**11*x**5) - log(d + e*x)/(6*d*e**6), True))/d**6 - b*e**6* Piecewise((1/(e**7*x), Eq(d, 0)), (-1/(6*d*(d/x + e)**6), True))*log(c*x** n)/d**6 - 6*b*e**5*n*Piecewise((-1/(e**6*x), Eq(d, 0)), (-25*d**3/(60*d**4 *e**5 + 240*d**3*e**6*x + 360*d**2*e**7*x**2 + 240*d*e**8*x**3 + 60*e**...
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x} \,d x } \]
1/60*a*((60*e^5*x^5 + 330*d*e^4*x^4 + 740*d^2*e^3*x^3 + 855*d^3*e^2*x^2 + 522*d^4*e*x + 147*d^5)/(d^6*e^6*x^6 + 6*d^7*e^5*x^5 + 15*d^8*e^4*x^4 + 20* d^9*e^3*x^3 + 15*d^10*e^2*x^2 + 6*d^11*e*x + d^12) - 60*log(e*x + d)/d^7 + 60*log(x)/d^7) + b*integrate((log(c) + log(x^n))/(e^7*x^8 + 7*d*e^6*x^7 + 21*d^2*e^5*x^6 + 35*d^3*e^4*x^5 + 35*d^4*e^3*x^4 + 21*d^5*e^2*x^3 + 7*d^6 *e*x^2 + d^7*x), x)
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x} \,d x } \]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x\right )}^7} \,d x \]