Integrand size = 21, antiderivative size = 229 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {10 b d n x}{e^5}-\frac {d (60 a+47 b n) x}{6 e^5}-\frac {5 b n x^2}{2 e^4}-\frac {10 b d x \log \left (c x^n\right )}{e^5}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^4 \left (5 a+b n+5 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^3 \left (20 a+9 b n+20 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}+\frac {x^2 \left (60 a+47 b n+60 b \log \left (c x^n\right )\right )}{12 e^4}+\frac {d^2 \left (60 a+47 b n+60 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{6 e^6}+\frac {10 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^6} \]
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Time = 0.31 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2384, 45, 2393, 2332, 2341, 2354, 2438} \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {d^2 \log \left (\frac {e x}{d}+1\right ) \left (60 a+60 b \log \left (c x^n\right )+47 b n\right )}{6 e^6}-\frac {x^3 \left (20 a+20 b \log \left (c x^n\right )+9 b n\right )}{6 e^3 (d+e x)}-\frac {x^4 \left (5 a+5 b \log \left (c x^n\right )+b n\right )}{6 e^2 (d+e x)^2}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}+\frac {x^2 \left (60 a+60 b \log \left (c x^n\right )+47 b n\right )}{12 e^4}-\frac {d x (60 a+47 b n)}{6 e^5}-\frac {10 b d x \log \left (c x^n\right )}{e^5}+\frac {10 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^6}+\frac {10 b d n x}{e^5}-\frac {5 b n x^2}{2 e^4} \]
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Rule 45
Rule 2332
Rule 2341
Rule 2354
Rule 2384
Rule 2393
Rule 2438
Rubi steps \begin{align*} \text {integral}& = -\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}+\frac {\int \frac {x^4 \left (5 a+b n+5 b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx}{3 e} \\ & = -\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^4 \left (5 a+b n+5 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}+\frac {\int \frac {x^3 \left (5 b n+4 (5 a+b n)+20 b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx}{6 e^2} \\ & = -\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^4 \left (5 a+b n+5 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^3 \left (20 a+9 b n+20 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}+\frac {\int \frac {x^2 \left (20 b n+3 (5 b n+4 (5 a+b n))+60 b \log \left (c x^n\right )\right )}{d+e x} \, dx}{6 e^3} \\ & = -\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^4 \left (5 a+b n+5 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^3 \left (20 a+9 b n+20 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}+\frac {\int \left (-\frac {d \left (20 b n+3 (5 b n+4 (5 a+b n))+60 b \log \left (c x^n\right )\right )}{e^2}+\frac {x \left (20 b n+3 (5 b n+4 (5 a+b n))+60 b \log \left (c x^n\right )\right )}{e}+\frac {d^2 \left (20 b n+3 (5 b n+4 (5 a+b n))+60 b \log \left (c x^n\right )\right )}{e^2 (d+e x)}\right ) \, dx}{6 e^3} \\ & = -\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^4 \left (5 a+b n+5 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^3 \left (20 a+9 b n+20 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}-\frac {d \int \left (20 b n+3 (5 b n+4 (5 a+b n))+60 b \log \left (c x^n\right )\right ) \, dx}{6 e^5}+\frac {d^2 \int \frac {20 b n+3 (5 b n+4 (5 a+b n))+60 b \log \left (c x^n\right )}{d+e x} \, dx}{6 e^5}+\frac {\int x \left (20 b n+3 (5 b n+4 (5 a+b n))+60 b \log \left (c x^n\right )\right ) \, dx}{6 e^4} \\ & = -\frac {d (60 a+47 b n) x}{6 e^5}-\frac {5 b n x^2}{2 e^4}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^4 \left (5 a+b n+5 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^3 \left (20 a+9 b n+20 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}+\frac {x^2 \left (60 a+47 b n+60 b \log \left (c x^n\right )\right )}{12 e^4}+\frac {d^2 \left (60 a+47 b n+60 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{6 e^6}-\frac {(10 b d) \int \log \left (c x^n\right ) \, dx}{e^5}-\frac {\left (10 b d^2 n\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^6} \\ & = \frac {10 b d n x}{e^5}-\frac {d (60 a+47 b n) x}{6 e^5}-\frac {5 b n x^2}{2 e^4}-\frac {10 b d x \log \left (c x^n\right )}{e^5}-\frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^4 \left (5 a+b n+5 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^3 \left (20 a+9 b n+20 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}+\frac {x^2 \left (60 a+47 b n+60 b \log \left (c x^n\right )\right )}{12 e^4}+\frac {d^2 \left (60 a+47 b n+60 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{6 e^6}+\frac {10 b d^2 n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^6} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.09 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {-48 a d e x+48 b d e n x-3 b e^2 n x^2-48 b d e x \log \left (c x^n\right )+6 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )+\frac {4 d^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}-\frac {30 d^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac {120 d^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x}-2 b d^2 n \left (\frac {d (3 d+2 e x)}{(d+e x)^2}+2 \log (x)-2 \log (d+e x)\right )-120 b d^2 n (\log (x)-\log (d+e x))+30 b d^2 n \left (\frac {d}{d+e x}+\log (x)-\log (d+e x)\right )+120 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+120 b d^2 n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{12 e^6} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.56 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.77
method | result | size |
risch | \(\frac {b \ln \left (x^{n}\right ) x^{2}}{2 e^{4}}-\frac {4 b \ln \left (x^{n}\right ) d x}{e^{5}}+\frac {b \ln \left (x^{n}\right ) d^{5}}{3 e^{6} \left (e x +d \right )^{3}}+\frac {10 b \ln \left (x^{n}\right ) d^{2} \ln \left (e x +d \right )}{e^{6}}+\frac {10 b \ln \left (x^{n}\right ) d^{3}}{e^{6} \left (e x +d \right )}-\frac {5 b \ln \left (x^{n}\right ) d^{4}}{2 e^{6} \left (e x +d \right )^{2}}-\frac {b n \,x^{2}}{4 e^{4}}+\frac {4 b d n x}{e^{5}}+\frac {17 b n \,d^{2}}{4 e^{6}}+\frac {47 b n \,d^{2} \ln \left (e x +d \right )}{6 e^{6}}+\frac {13 b n \,d^{3}}{6 e^{6} \left (e x +d \right )}-\frac {b n \,d^{4}}{6 e^{6} \left (e x +d \right )^{2}}-\frac {47 b n \,d^{2} \ln \left (e x \right )}{6 e^{6}}-\frac {10 b n \,d^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{6}}-\frac {10 b n \,d^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{6}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {\frac {1}{2} e \,x^{2}-4 d x}{e^{5}}+\frac {d^{5}}{3 e^{6} \left (e x +d \right )^{3}}+\frac {10 d^{2} \ln \left (e x +d \right )}{e^{6}}+\frac {10 d^{3}}{e^{6} \left (e x +d \right )}-\frac {5 d^{4}}{2 e^{6} \left (e x +d \right )^{2}}\right )\) | \(405\) |
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\[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{5}}{{\left (e x + d\right )}^{4}} \,d x } \]
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Time = 62.66 (sec) , antiderivative size = 617, normalized size of antiderivative = 2.69 \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\text {Too large to display} \]
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\[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{5}}{{\left (e x + d\right )}^{4}} \,d x } \]
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\[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{5}}{{\left (e x + d\right )}^{4}} \,d x } \]
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Timed out. \[ \int \frac {x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\int \frac {x^5\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{{\left (d+e\,x\right )}^4} \,d x \]
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