Integrand size = 21, antiderivative size = 401 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=-\frac {b n}{4 d^7 x^2}+\frac {7 b e n}{d^8 x}-\frac {b e^2 n}{30 d^4 (d+e x)^5}-\frac {23 b e^2 n}{120 d^5 (d+e x)^4}-\frac {34 b e^2 n}{45 d^6 (d+e x)^3}-\frac {14 b e^2 n}{5 d^7 (d+e x)^2}-\frac {131 b e^2 n}{10 d^8 (d+e x)}-\frac {131 b e^2 n \log (x)}{10 d^9}-\frac {a+b \log \left (c x^n\right )}{2 d^7 x^2}+\frac {7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac {15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}-\frac {21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}-\frac {28 e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^9}+\frac {341 b e^2 n \log (d+e x)}{10 d^9}+\frac {28 b e^2 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^9} \]
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Time = 0.43 (sec) , antiderivative size = 401, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {46, 2393, 2341, 2356, 2351, 31, 2379, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=-\frac {21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}-\frac {28 e^2 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^9}+\frac {7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}+\frac {15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}-\frac {a+b \log \left (c x^n\right )}{2 d^7 x^2}+\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}+\frac {28 b e^2 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^9}-\frac {131 b e^2 n \log (x)}{10 d^9}+\frac {341 b e^2 n \log (d+e x)}{10 d^9}-\frac {131 b e^2 n}{10 d^8 (d+e x)}+\frac {7 b e n}{d^8 x}-\frac {14 b e^2 n}{5 d^7 (d+e x)^2}-\frac {b n}{4 d^7 x^2}-\frac {34 b e^2 n}{45 d^6 (d+e x)^3}-\frac {23 b e^2 n}{120 d^5 (d+e x)^4}-\frac {b e^2 n}{30 d^4 (d+e x)^5} \]
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Rule 31
Rule 46
Rule 2341
Rule 2351
Rule 2356
Rule 2379
Rule 2393
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a+b \log \left (c x^n\right )}{d^7 x^3}-\frac {7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x^2}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^7}-\frac {3 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)^6}-\frac {6 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)^5}-\frac {10 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)^4}-\frac {15 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^7 (d+e x)^3}-\frac {21 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^8 (d+e x)^2}+\frac {28 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^8 x (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{d^7}-\frac {(7 e) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^8}+\frac {\left (28 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^8}-\frac {\left (21 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^8}-\frac {\left (15 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^7}-\frac {\left (10 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d^6}-\frac {\left (6 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^5} \, dx}{d^5}-\frac {\left (3 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^6} \, dx}{d^4}-\frac {e^3 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^7} \, dx}{d^3} \\ & = -\frac {b n}{4 d^7 x^2}+\frac {7 b e n}{d^8 x}-\frac {a+b \log \left (c x^n\right )}{2 d^7 x^2}+\frac {7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac {15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}-\frac {21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}-\frac {28 e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^9}+\frac {\left (28 b e^2 n\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{d^9}-\frac {\left (15 b e^2 n\right ) \int \frac {1}{x (d+e x)^2} \, dx}{2 d^7}-\frac {\left (10 b e^2 n\right ) \int \frac {1}{x (d+e x)^3} \, dx}{3 d^6}-\frac {\left (3 b e^2 n\right ) \int \frac {1}{x (d+e x)^4} \, dx}{2 d^5}-\frac {\left (3 b e^2 n\right ) \int \frac {1}{x (d+e x)^5} \, dx}{5 d^4}-\frac {\left (b e^2 n\right ) \int \frac {1}{x (d+e x)^6} \, dx}{6 d^3}+\frac {\left (21 b e^3 n\right ) \int \frac {1}{d+e x} \, dx}{d^9} \\ & = -\frac {b n}{4 d^7 x^2}+\frac {7 b e n}{d^8 x}-\frac {a+b \log \left (c x^n\right )}{2 d^7 x^2}+\frac {7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac {15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}-\frac {21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}-\frac {28 e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^9}+\frac {21 b e^2 n \log (d+e x)}{d^9}+\frac {28 b e^2 n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^9}-\frac {\left (15 b e^2 n\right ) \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^7}-\frac {\left (10 b e^2 n\right ) \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^6}-\frac {\left (3 b e^2 n\right ) \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}{2 d^5}-\frac {\left (3 b e^2 n\right ) \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^4}-\frac {\left (b e^2 n\right ) \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^3} \\ & = -\frac {b n}{4 d^7 x^2}+\frac {7 b e n}{d^8 x}-\frac {b e^2 n}{30 d^4 (d+e x)^5}-\frac {23 b e^2 n}{120 d^5 (d+e x)^4}-\frac {34 b e^2 n}{45 d^6 (d+e x)^3}-\frac {14 b e^2 n}{5 d^7 (d+e x)^2}-\frac {131 b e^2 n}{10 d^8 (d+e x)}-\frac {131 b e^2 n \log (x)}{10 d^9}-\frac {a+b \log \left (c x^n\right )}{2 d^7 x^2}+\frac {7 e \left (a+b \log \left (c x^n\right )\right )}{d^8 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{6 d^3 (d+e x)^6}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{5 d^4 (d+e x)^5}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^5 (d+e x)^4}+\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^6 (d+e x)^3}+\frac {15 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^7 (d+e x)^2}-\frac {21 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^9 (d+e x)}-\frac {28 e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^9}+\frac {341 b e^2 n \log (d+e x)}{10 d^9}+\frac {28 b e^2 n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^9} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.21 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\frac {-\frac {180 a d^2}{x^2}-\frac {90 b d^2 n}{x^2}+\frac {2520 a d e}{x}+\frac {2520 b d e n}{x}+\frac {60 a d^6 e^2}{(d+e x)^6}+\frac {216 a d^5 e^2}{(d+e x)^5}-\frac {12 b d^5 e^2 n}{(d+e x)^5}+\frac {540 a d^4 e^2}{(d+e x)^4}-\frac {69 b d^4 e^2 n}{(d+e x)^4}+\frac {1200 a d^3 e^2}{(d+e x)^3}-\frac {272 b d^3 e^2 n}{(d+e x)^3}+\frac {2700 a d^2 e^2}{(d+e x)^2}-\frac {1008 b d^2 e^2 n}{(d+e x)^2}+\frac {7560 a d e^2}{d+e x}-\frac {4716 b d e^2 n}{d+e x}-12276 b e^2 n \log (x)+\frac {10080 a e^2 \log \left (c x^n\right )}{n}-\frac {180 b d^2 \log \left (c x^n\right )}{x^2}+\frac {2520 b d e \log \left (c x^n\right )}{x}+\frac {60 b d^6 e^2 \log \left (c x^n\right )}{(d+e x)^6}+\frac {216 b d^5 e^2 \log \left (c x^n\right )}{(d+e x)^5}+\frac {540 b d^4 e^2 \log \left (c x^n\right )}{(d+e x)^4}+\frac {1200 b d^3 e^2 \log \left (c x^n\right )}{(d+e x)^3}+\frac {2700 b d^2 e^2 \log \left (c x^n\right )}{(d+e x)^2}+\frac {7560 b d e^2 \log \left (c x^n\right )}{d+e x}+\frac {5040 b e^2 \log ^2\left (c x^n\right )}{n}+12276 b e^2 n \log (d+e x)-10080 a e^2 \log \left (1+\frac {e x}{d}\right )-10080 b e^2 \log \left (c x^n\right ) \log \left (1+\frac {e x}{d}\right )-10080 b e^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{360 d^9} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.62 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.48
method | result | size |
risch | \(-\frac {28 b \ln \left (x^{n}\right ) e^{2} \ln \left (e x +d \right )}{d^{9}}+\frac {21 b \ln \left (x^{n}\right ) e^{2}}{d^{8} \left (e x +d \right )}+\frac {15 b \ln \left (x^{n}\right ) e^{2}}{2 d^{7} \left (e x +d \right )^{2}}+\frac {10 b \ln \left (x^{n}\right ) e^{2}}{3 d^{6} \left (e x +d \right )^{3}}+\frac {3 b \ln \left (x^{n}\right ) e^{2}}{2 d^{5} \left (e x +d \right )^{4}}+\frac {3 b \ln \left (x^{n}\right ) e^{2}}{5 d^{4} \left (e x +d \right )^{5}}+\frac {b \ln \left (x^{n}\right ) e^{2}}{6 d^{3} \left (e x +d \right )^{6}}-\frac {b \ln \left (x^{n}\right )}{2 d^{7} x^{2}}+\frac {28 b \ln \left (x^{n}\right ) e^{2} \ln \left (x \right )}{d^{9}}+\frac {7 b \ln \left (x^{n}\right ) e}{d^{8} x}-\frac {131 b \,e^{2} n}{10 d^{8} \left (e x +d \right )}+\frac {341 b \,e^{2} n \ln \left (e x +d \right )}{10 d^{9}}-\frac {14 b \,e^{2} n}{5 d^{7} \left (e x +d \right )^{2}}-\frac {34 b \,e^{2} n}{45 d^{6} \left (e x +d \right )^{3}}-\frac {23 b \,e^{2} n}{120 d^{5} \left (e x +d \right )^{4}}-\frac {b \,e^{2} n}{30 d^{4} \left (e x +d \right )^{5}}-\frac {b n}{4 d^{7} x^{2}}+\frac {7 b e n}{d^{8} x}-\frac {341 b \,e^{2} n \ln \left (x \right )}{10 d^{9}}-\frac {14 b n \,e^{2} \ln \left (x \right )^{2}}{d^{9}}+\frac {28 b n \,e^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{9}}+\frac {28 b n \,e^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{9}}+\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 {28 e^{2} \ln \left (e x +d \right )}{d^{9}}+\frac {21 e^{2}}{d^{8} \left (e x +d \right )}+\frac {15 e^{2}}{2 d^{7} \left (e x +d \right )^{2}}+\frac {10 e^{2}}{3 d^{6} \left (e x +d \right )^{3}}+\frac {3 e^{2}}{2 d^{5} \left (e x +d \right )^{4}}+\frac {3 e^{2}}{5 d^{4} \left (e x +d \right )^{5}}+\frac {e^{2}}{6 d^{3} \left (e x +d \right )^{6}}-\frac {1}{2 d^{7} x^{2}}+\frac {28 e^{2} \ln \left (x \right )}{d^{9}}+\frac {7 e}{d^{8} x}\right )\) | \(594\) |
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{3}} \,d x } \]
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Time = 156.80 (sec) , antiderivative size = 1737, normalized size of antiderivative = 4.33 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\text {Too large to display} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{3}} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{7} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^7} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,{\left (d+e\,x\right )}^7} \,d x \]
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