Integrand size = 21, antiderivative size = 263 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^4} \, dx=-\frac {b n}{4 d^4 x^2}+\frac {4 b e n}{d^5 x}-\frac {b e^2 n}{6 d^4 (d+e x)^2}-\frac {11 b e^2 n}{6 d^5 (d+e x)}-\frac {11 b e^2 n \log (x)}{6 d^6}-\frac {a+b \log \left (c x^n\right )}{2 d^4 x^2}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac {6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}-\frac {10 e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^6}+\frac {47 b e^2 n \log (d+e x)}{6 d^6}+\frac {10 b e^2 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^6} \]
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Time = 0.25 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 14, 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} \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^4} \, dx=-\frac {6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}-\frac {10 e^2 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^6}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac {a+b \log \left (c x^n\right )}{2 d^4 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}+\frac {10 b e^2 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^6}-\frac {11 b e^2 n \log (x)}{6 d^6}+\frac {47 b e^2 n \log (d+e x)}{6 d^6}-\frac {11 b e^2 n}{6 d^5 (d+e x)}+\frac {4 b e n}{d^5 x}-\frac {b e^2 n}{6 d^4 (d+e x)^2}-\frac {b n}{4 d^4 x^2} \]
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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^4 x^3}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x^2}-\frac {e^3 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^4}-\frac {3 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)^3}-\frac {6 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)^2}+\frac {10 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^5 x (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{d^4}-\frac {(4 e) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^5}+\frac {\left (10 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^5}-\frac {\left (6 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^5}-\frac {\left (3 e^3\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^4}-\frac {e^3 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d^3} \\ & = -\frac {b n}{4 d^4 x^2}+\frac {4 b e n}{d^5 x}-\frac {a+b \log \left (c x^n\right )}{2 d^4 x^2}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac {6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}-\frac {10 e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^6}+\frac {\left (10 b e^2 n\right ) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{d^6}-\frac {\left (3 b e^2 n\right ) \int \frac {1}{x (d+e x)^2} \, dx}{2 d^4}-\frac {\left (b e^2 n\right ) \int \frac {1}{x (d+e x)^3} \, dx}{3 d^3}+\frac {\left (6 b e^3 n\right ) \int \frac {1}{d+e x} \, dx}{d^6} \\ & = -\frac {b n}{4 d^4 x^2}+\frac {4 b e n}{d^5 x}-\frac {a+b \log \left (c x^n\right )}{2 d^4 x^2}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac {6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}-\frac {10 e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^6}+\frac {6 b e^2 n \log (d+e x)}{d^6}+\frac {10 b e^2 n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^6}-\frac {\left (3 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^4}-\frac {\left (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^3} \\ & = -\frac {b n}{4 d^4 x^2}+\frac {4 b e n}{d^5 x}-\frac {b e^2 n}{6 d^4 (d+e x)^2}-\frac {11 b e^2 n}{6 d^5 (d+e x)}-\frac {11 b e^2 n \log (x)}{6 d^6}-\frac {a+b \log \left (c x^n\right )}{2 d^4 x^2}+\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac {6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}-\frac {10 e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^6}+\frac {47 b e^2 n \log (d+e x)}{6 d^6}+\frac {10 b e^2 n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^6} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.05 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^4} \, dx=\frac {-\frac {3 b d^2 n}{x^2}+\frac {48 b d e n}{x}-\frac {18 b d e^2 n}{d+e x}-\frac {2 b d e^2 n (3 d+2 e x)}{(d+e x)^2}-22 b e^2 n \log (x)-\frac {6 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac {48 d e \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {4 d^3 e^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}+\frac {18 d^2 e^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac {72 d e^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x}+\frac {60 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}-72 b e^2 n (\log (x)-\log (d+e x))+22 b e^2 n \log (d+e x)-120 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-120 b e^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{12 d^6} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.92 (sec) , antiderivative size = 438, normalized size of antiderivative = 1.67
method | result | size |
risch | \(-\frac {10 b \ln \left (x^{n}\right ) e^{2} \ln \left (e x +d \right )}{d^{6}}+\frac {6 b \ln \left (x^{n}\right ) e^{2}}{d^{5} \left (e x +d \right )}+\frac {3 b \ln \left (x^{n}\right ) e^{2}}{2 d^{4} \left (e x +d \right )^{2}}+\frac {b \ln \left (x^{n}\right ) e^{2}}{3 d^{3} \left (e x +d \right )^{3}}-\frac {b \ln \left (x^{n}\right )}{2 d^{4} x^{2}}+\frac {10 b \ln \left (x^{n}\right ) e^{2} \ln \left (x \right )}{d^{6}}+\frac {4 b \ln \left (x^{n}\right ) e}{d^{5} x}-\frac {11 b \,e^{2} n}{6 d^{5} \left (e x +d \right )}+\frac {47 b \,e^{2} n \ln \left (e x +d \right )}{6 d^{6}}-\frac {b \,e^{2} n}{6 d^{4} \left (e x +d \right )^{2}}-\frac {b n}{4 d^{4} x^{2}}+\frac {4 b e n}{d^{5} x}-\frac {47 b \,e^{2} n \ln \left (x \right )}{6 d^{6}}-\frac {5 b n \,e^{2} \ln \left (x \right )^{2}}{d^{6}}+\frac {10 b n \,e^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{6}}+\frac {10 b n \,e^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{6}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {10 e^{2} \ln \left (e x +d \right )}{d^{6}}+\frac {6 e^{2}}{d^{5} \left (e x +d \right )}+\frac {3 e^{2}}{2 d^{4} \left (e x +d \right )^{2}}+\frac {e^{2}}{3 d^{3} \left (e x +d \right )^{3}}-\frac {1}{2 d^{4} x^{2}}+\frac {10 e^{2} \ln \left (x \right )}{d^{6}}+\frac {4 e}{d^{5} x}\right )\) | \(438\) |
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^4} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{4} x^{3}} \,d x } \]
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Time = 61.94 (sec) , antiderivative size = 668, normalized size of antiderivative = 2.54 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^4} \, dx=\text {Too large to display} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^4} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{4} x^{3}} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^4} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{4} x^{3}} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 (d+e x)^4} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,{\left (d+e\,x\right )}^4} \,d x \]
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