Integrand size = 21, antiderivative size = 294 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=-\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {29 b n}{20 d^6 (d+e x)}-\frac {29 b n \log (x)}{20 d^7}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}-\frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^7}+\frac {49 b n \log (d+e x)}{20 d^7}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^7} \]
[Out]
Time = 0.47 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2389, 2379, 2438, 2351, 31, 2356, 46} \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=-\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^7}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)}+\frac {a+b \log \left (c x^n\right )}{2 d^5 (d+e x)^2}+\frac {a+b \log \left (c x^n\right )}{3 d^4 (d+e x)^3}+\frac {a+b \log \left (c x^n\right )}{4 d^3 (d+e x)^4}+\frac {a+b \log \left (c x^n\right )}{5 d^2 (d+e x)^5}+\frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^7}+\frac {49 b n \log (d+e x)}{20 d^7}-\frac {29 b n \log (x)}{20 d^7}-\frac {29 b n}{20 d^6 (d+e x)}-\frac {19 b n}{40 d^5 (d+e x)^2}-\frac {37 b n}{180 d^4 (d+e x)^3}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {b n}{30 d^2 (d+e x)^5} \]
[In]
[Out]
Rule 31
Rule 46
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \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} \\ & = \frac {a+b \log \left (c x^n\right )}{6 d (d+e x)^6}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^5} \, dx}{d^2}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{d^2}-\frac {(b n) \int \frac {1}{x (d+e x)^6} \, dx}{6 d} \\ & = \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 {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^4} \, dx}{d^3}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{d^3}-\frac {(b n) \int \frac {1}{x (d+e x)^5} \, dx}{5 d^2}-\frac {(b n) \int \left (\frac {1}{d^6 x}-\frac {e}{d (d+e x)^6}-\frac {e}{d^2 (d+e x)^5}-\frac {e}{d^3 (d+e x)^4}-\frac {e}{d^4 (d+e x)^3}-\frac {e}{d^5 (d+e x)^2}-\frac {e}{d^6 (d+e x)}\right ) \, dx}{6 d} \\ & = -\frac {b n}{30 d^2 (d+e x)^5}-\frac {b n}{24 d^3 (d+e x)^4}-\frac {b n}{18 d^4 (d+e x)^3}-\frac {b n}{12 d^5 (d+e x)^2}-\frac {b n}{6 d^6 (d+e x)}-\frac {b n \log (x)}{6 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 {b n \log (d+e x)}{6 d^7}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^3} \, dx}{d^4}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d^4}-\frac {(b n) \int \frac {1}{x (d+e x)^4} \, dx}{4 d^3}-\frac {(b n) \int \left (\frac {1}{d^5 x}-\frac {e}{d (d+e x)^5}-\frac {e}{d^2 (d+e x)^4}-\frac {e}{d^3 (d+e x)^3}-\frac {e}{d^4 (d+e x)^2}-\frac {e}{d^5 (d+e x)}\right ) \, dx}{5 d^2} \\ & = -\frac {b n}{30 d^2 (d+e x)^5}-\frac {11 b n}{120 d^3 (d+e x)^4}-\frac {11 b n}{90 d^4 (d+e x)^3}-\frac {11 b n}{60 d^5 (d+e x)^2}-\frac {11 b n}{30 d^6 (d+e x)}-\frac {11 b n \log (x)}{30 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 {11 b n \log (d+e x)}{30 d^7}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx}{d^5}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^5}-\frac {(b n) \int \frac {1}{x (d+e x)^3} \, dx}{3 d^4}-\frac {(b n) \int \left (\frac {1}{d^4 x}-\frac {e}{d (d+e x)^4}-\frac {e}{d^2 (d+e x)^3}-\frac {e}{d^3 (d+e x)^2}-\frac {e}{d^4 (d+e x)}\right ) \, dx}{4 d^3} \\ & = -\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 {37 b n}{120 d^5 (d+e x)^2}-\frac {37 b n}{60 d^6 (d+e x)}-\frac {37 b n \log (x)}{60 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 {37 b n \log (d+e x)}{60 d^7}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^6}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^6}-\frac {(b n) \int \frac {1}{x (d+e x)^2} \, dx}{2 d^5}-\frac {(b n) \int \left (\frac {1}{d^3 x}-\frac {e}{d (d+e x)^3}-\frac {e}{d^2 (d+e x)^2}-\frac {e}{d^3 (d+e x)}\right ) \, dx}{3 d^4} \\ & = -\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 {19 b n}{20 d^6 (d+e x)}-\frac {19 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 {19 b n \log (d+e x)}{20 d^7}+\frac {(b n) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{d^7}-\frac {(b n) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{2 d^5}+\frac {(b e n) \int \frac {1}{d+e x} \, dx}{d^7} \\ & = -\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 \text {Li}_2\left (-\frac {d}{e x}\right )}{d^7} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\frac {\frac {60 a d^6}{(d+e x)^6}+\frac {72 a d^5}{(d+e x)^5}-\frac {12 b d^5 n}{(d+e x)^5}+\frac {90 a d^4}{(d+e x)^4}-\frac {33 b d^4 n}{(d+e x)^4}+\frac {120 a d^3}{(d+e x)^3}-\frac {74 b d^3 n}{(d+e x)^3}+\frac {180 a d^2}{(d+e x)^2}-\frac {171 b d^2 n}{(d+e x)^2}+\frac {360 a d}{d+e x}-\frac {522 b d n}{d+e x}-882 b n \log (x)+\frac {360 a \log \left (c x^n\right )}{n}+\frac {60 b d^6 \log \left (c x^n\right )}{(d+e x)^6}+\frac {72 b d^5 \log \left (c x^n\right )}{(d+e x)^5}+\frac {90 b d^4 \log \left (c x^n\right )}{(d+e x)^4}+\frac {120 b d^3 \log \left (c x^n\right )}{(d+e x)^3}+\frac {180 b d^2 \log \left (c x^n\right )}{(d+e x)^2}+\frac {360 b d \log \left (c x^n\right )}{d+e x}+\frac {180 b \log ^2\left (c x^n\right )}{n}+882 b n \log (d+e x)-360 a \log \left (1+\frac {e x}{d}\right )-360 b \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )-360 b n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{360 d^7} \]
[In]
[Out]
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 | \(-\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{7}}+\frac {b \ln \left (x^{n}\right )}{d^{6} \left (e x +d \right )}+\frac {b \ln \left (x^{n}\right )}{2 d^{5} \left (e x +d \right )^{2}}+\frac {b \ln \left (x^{n}\right )}{3 d^{4} \left (e x +d \right )^{3}}+\frac {b \ln \left (x^{n}\right )}{4 d^{3} \left (e x +d \right )^{4}}+\frac {b \ln \left (x^{n}\right )}{5 d^{2} \left (e x +d \right )^{5}}+\frac {b \ln \left (x^{n}\right )}{6 d \left (e x +d \right )^{6}}+\frac {b \ln \left (x^{n}\right ) \ln \left (x \right )}{d^{7}}-\frac {29 b n}{20 d^{6} \left (e x +d \right )}-\frac {19 b n}{40 d^{5} \left (e x +d \right )^{2}}-\frac {37 b n}{180 d^{4} \left (e x +d \right )^{3}}-\frac {11 b n}{120 d^{3} \left (e x +d \right )^{4}}-\frac {b n}{30 d^{2} \left (e x +d \right )^{5}}+\frac {49 b n \ln \left (e x +d \right )}{20 d^{7}}-\frac {49 b n \ln \left (x \right )}{20 d^{7}}-\frac {b n \ln \left (x \right )^{2}}{2 d^{7}}+\frac {b n \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{7}}+\frac {b n \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{7}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (e x +d \right )}{d^{7}}+\frac {1}{d^{6} \left (e x +d \right )}+\frac {1}{2 d^{5} \left (e x +d \right )^{2}}+\frac {1}{3 d^{4} \left (e x +d \right )^{3}}+\frac {1}{4 d^{3} \left (e x +d \right )^{4}}+\frac {1}{5 d^{2} \left (e x +d \right )^{5}}+\frac {1}{6 d \left (e x +d \right )^{6}}+\frac {\ln \left (x \right )}{d^{7}}\right )\) | \(445\) |
[In]
[Out]
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x} \,d x } \]
[In]
[Out]
Time = 156.44 (sec) , antiderivative size = 1518, normalized size of antiderivative = 5.16 \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x} \,d x } \]
[In]
[Out]
\[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^7} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x\right )}^7} \,d x \]
[In]
[Out]