Integrand size = 23, antiderivative size = 169 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=-\frac {b n}{2 d^2 r^2 \left (d+e x^r\right )}-\frac {b n \log (x)}{2 d^3 r}+\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}+\frac {3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^3 r^2} \]
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Time = 0.30 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2391, 2379, 2438, 2373, 266, 2376, 272, 46} \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}+\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}+\frac {b n \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^3 r^2}+\frac {3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}-\frac {b n \log (x)}{2 d^3 r}-\frac {b n}{2 d^2 r^2 \left (d+e x^r\right )} \]
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Rule 46
Rule 266
Rule 272
Rule 2373
Rule 2376
Rule 2379
Rule 2391
Rule 2438
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^2} \, dx}{d}-\frac {e \int \frac {x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^3} \, dx}{d} \\ & = \frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx}{d^2}-\frac {e \int \frac {x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx}{d^2}-\frac {(b n) \int \frac {1}{x \left (d+e x^r\right )^2} \, dx}{2 d r} \\ & = \frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}-\frac {(b n) \text {Subst}\left (\int \frac {1}{x (d+e x)^2} \, dx,x,x^r\right )}{2 d r^2}+\frac {(b n) \int \frac {\log \left (1+\frac {d x^{-r}}{e}\right )}{x} \, dx}{d^3 r}+\frac {(b e n) \int \frac {x^{-1+r}}{d+e x^r} \, dx}{d^3 r} \\ & = \frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}+\frac {b n \log \left (d+e x^r\right )}{d^3 r^2}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^2}-\frac {(b n) \text {Subst}\left (\int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx,x,x^r\right )}{2 d r^2} \\ & = -\frac {b n}{2 d^2 r^2 \left (d+e x^r\right )}-\frac {b n \log (x)}{2 d^3 r}+\frac {a+b \log \left (c x^n\right )}{2 d r \left (d+e x^r\right )^2}-\frac {e x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r \left (d+e x^r\right )}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}+\frac {3 b n \log \left (d+e x^r\right )}{2 d^3 r^2}+\frac {b n \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^2} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.01 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\frac {\frac {d^2 r \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2}+\frac {d \left (-b n+2 a r+2 b r \log \left (c x^n\right )\right )}{d+e x^r}+3 b n \log \left (d-d x^r\right )-2 a r \log \left (d-d x^r\right )+2 b r \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+2 b n \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )\right )}{2 d^3 r^2} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.00 (sec) , antiderivative size = 473, normalized size of antiderivative = 2.80
method | result | size |
risch | \(\frac {b \ln \left (d +e \,x^{r}\right ) n \ln \left (x \right )}{r \,d^{3}}-\frac {b \ln \left (d +e \,x^{r}\right ) \ln \left (x^{n}\right )}{r \,d^{3}}-\frac {b n \ln \left (x \right )}{r \,d^{2} \left (d +e \,x^{r}\right )}+\frac {b \ln \left (x^{n}\right )}{r \,d^{2} \left (d +e \,x^{r}\right )}-\frac {b n \ln \left (x \right )}{2 r d \left (d +e \,x^{r}\right )^{2}}+\frac {b \ln \left (x^{n}\right )}{2 r d \left (d +e \,x^{r}\right )^{2}}-\frac {b \ln \left (x^{r}\right ) n \ln \left (x \right )}{r \,d^{3}}+\frac {b \ln \left (x^{r}\right ) \ln \left (x^{n}\right )}{r \,d^{3}}+\frac {3 b n \ln \left (d +e \,x^{r}\right )}{2 d^{3} r^{2}}-\frac {b n e \ln \left (x \right ) x^{r}}{r \,d^{3} \left (d +e \,x^{r}\right )}-\frac {b n \operatorname {dilog}\left (\frac {d +e \,x^{r}}{d}\right )}{r^{2} d^{3}}-\frac {b n \ln \left (x \right ) \ln \left (\frac {d +e \,x^{r}}{d}\right )}{r \,d^{3}}-\frac {b n}{2 d^{2} r^{2} \left (d +e \,x^{r}\right )}-\frac {b n \,e^{2} \ln \left (x \right ) x^{2 r}}{2 r \,d^{3} \left (d +e \,x^{r}\right )^{2}}-\frac {b n e \ln \left (x \right ) x^{r}}{r \,d^{2} \left (d +e \,x^{r}\right )^{2}}+\frac {b n \ln \left (x \right )^{2}}{2 d^{3}}+\frac {\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {\ln \left (d +e \,x^{r}\right )}{d^{3}}+\frac {1}{d^{2} \left (d +e \,x^{r}\right )}+\frac {1}{2 d \left (d +e \,x^{r}\right )^{2}}+\frac {\ln \left (x^{r}\right )}{d^{3}}\right )}{r}\) | \(473\) |
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Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (160) = 320\).
Time = 0.28 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.37 \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\frac {b d^{2} n r^{2} \log \left (x\right )^{2} + 3 \, b d^{2} r \log \left (c\right ) - b d^{2} n + 3 \, a d^{2} r + {\left (b e^{2} n r^{2} \log \left (x\right )^{2} + {\left (2 \, b e^{2} r^{2} \log \left (c\right ) - 3 \, b e^{2} n r + 2 \, a e^{2} r^{2}\right )} \log \left (x\right )\right )} x^{2 \, r} + {\left (2 \, b d e n r^{2} \log \left (x\right )^{2} + 2 \, b d e r \log \left (c\right ) - b d e n + 2 \, a d e r + 4 \, {\left (b d e r^{2} \log \left (c\right ) - b d e n r + a d e r^{2}\right )} \log \left (x\right )\right )} x^{r} - 2 \, {\left (b e^{2} n x^{2 \, r} + 2 \, b d e n x^{r} + b d^{2} n\right )} {\rm Li}_2\left (-\frac {e x^{r} + d}{d} + 1\right ) - {\left (2 \, b d^{2} r \log \left (c\right ) - 3 \, b d^{2} n + 2 \, a d^{2} r + {\left (2 \, b e^{2} r \log \left (c\right ) - 3 \, b e^{2} n + 2 \, a e^{2} r\right )} x^{2 \, r} + 2 \, {\left (2 \, b d e r \log \left (c\right ) - 3 \, b d e n + 2 \, a d e r\right )} x^{r}\right )} \log \left (e x^{r} + d\right ) + 2 \, {\left (b d^{2} r^{2} \log \left (c\right ) + a d^{2} r^{2}\right )} \log \left (x\right ) - 2 \, {\left (b e^{2} n r x^{2 \, r} \log \left (x\right ) + 2 \, b d e n r x^{r} \log \left (x\right ) + b d^{2} n r \log \left (x\right )\right )} \log \left (\frac {e x^{r} + d}{d}\right )}{2 \, {\left (d^{3} e^{2} r^{2} x^{2 \, r} + 2 \, d^{4} e r^{2} x^{r} + d^{5} r^{2}\right )}} \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\text {Timed out} \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{3} x} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{r} + d\right )}^{3} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^3} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x^r\right )}^3} \,d x \]
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