Integrand size = 24, antiderivative size = 238 \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )}{d}+\frac {6 n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {6 n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,1+\frac {b x}{a}\right )}{d}-\frac {6 n^3 \operatorname {PolyLog}\left (4,-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {6 n^3 \operatorname {PolyLog}\left (4,1+\frac {b x}{a}\right )}{d} \]
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Time = 0.26 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1607, 2463, 2443, 2481, 2421, 2430, 6724} \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\frac {6 n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {6 n^2 \operatorname {PolyLog}\left (3,\frac {b x}{a}+1\right ) \log \left (c (a+b x)^n\right )}{d}+\frac {3 n \operatorname {PolyLog}\left (2,\frac {b x}{a}+1\right ) \log ^2\left (c (a+b x)^n\right )}{d}+\frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {6 n^3 \operatorname {PolyLog}\left (4,-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {6 n^3 \operatorname {PolyLog}\left (4,\frac {b x}{a}+1\right )}{d} \]
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Rule 1607
Rule 2421
Rule 2430
Rule 2443
Rule 2463
Rule 2481
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \frac {\log ^3\left (c (a+b x)^n\right )}{x (d+e x)} \, dx \\ & = \int \left (\frac {\log ^3\left (c (a+b x)^n\right )}{d x}-\frac {e \log ^3\left (c (a+b x)^n\right )}{d (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {\log ^3\left (c (a+b x)^n\right )}{x} \, dx}{d}-\frac {e \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x} \, dx}{d} \\ & = \frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {(3 b n) \int \frac {\log \left (-\frac {b x}{a}\right ) \log ^2\left (c (a+b x)^n\right )}{a+b x} \, dx}{d}+\frac {(3 b n) \int \frac {\log ^2\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{a+b x} \, dx}{d} \\ & = \frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {(3 n) \text {Subst}\left (\int \frac {\log ^2\left (c x^n\right ) \log \left (-\frac {b \left (-\frac {a}{b}+\frac {x}{b}\right )}{a}\right )}{x} \, dx,x,a+b x\right )}{d}+\frac {(3 n) \text {Subst}\left (\int \frac {\log ^2\left (c x^n\right ) \log \left (\frac {b \left (\frac {b d-a e}{b}+\frac {e x}{b}\right )}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d} \\ & = \frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (1+\frac {b x}{a}\right )}{d}-\frac {\left (6 n^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^n\right ) \text {Li}_2\left (\frac {x}{a}\right )}{x} \, dx,x,a+b x\right )}{d}+\frac {\left (6 n^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^n\right ) \text {Li}_2\left (-\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d} \\ & = \frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (1+\frac {b x}{a}\right )}{d}+\frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (1+\frac {b x}{a}\right )}{d}+\frac {\left (6 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{a}\right )}{x} \, dx,x,a+b x\right )}{d}-\frac {\left (6 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {e x}{b d-a e}\right )}{x} \, dx,x,a+b x\right )}{d} \\ & = \frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (1+\frac {b x}{a}\right )}{d}+\frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (1+\frac {b x}{a}\right )}{d}-\frac {6 n^3 \text {Li}_4\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {6 n^3 \text {Li}_4\left (1+\frac {b x}{a}\right )}{d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(494\) vs. \(2(238)=476\).
Time = 0.14 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.08 \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\frac {-\log (x) \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right )^3+\left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right )^3 \log (d+e x)+3 n \left (-n \log (a+b x)+\log \left (c (a+b x)^n\right )\right )^2 \left (\log (x) \left (\log (a+b x)-\log \left (1+\frac {b x}{a}\right )\right )-\log (a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-\operatorname {PolyLog}\left (2,-\frac {b x}{a}\right )-\operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )\right )-3 n^2 \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right ) \left (\log \left (-\frac {b x}{a}\right ) \log ^2(a+b x)-\log ^2(a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-2 \log (a+b x) \operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )+2 \log (a+b x) \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )+2 \operatorname {PolyLog}\left (3,\frac {e (a+b x)}{-b d+a e}\right )-2 \operatorname {PolyLog}\left (3,1+\frac {b x}{a}\right )\right )+n^3 \left (\log \left (-\frac {b x}{a}\right ) \log ^3(a+b x)-\log ^3(a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-3 \log ^2(a+b x) \operatorname {PolyLog}\left (2,\frac {e (a+b x)}{-b d+a e}\right )+3 \log ^2(a+b x) \operatorname {PolyLog}\left (2,1+\frac {b x}{a}\right )+6 \log (a+b x) \operatorname {PolyLog}\left (3,\frac {e (a+b x)}{-b d+a e}\right )-6 \log (a+b x) \operatorname {PolyLog}\left (3,1+\frac {b x}{a}\right )-6 \operatorname {PolyLog}\left (4,\frac {e (a+b x)}{-b d+a e}\right )+6 \operatorname {PolyLog}\left (4,1+\frac {b x}{a}\right )\right )}{d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.02 (sec) , antiderivative size = 1756, normalized size of antiderivative = 7.38
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\[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d x} \,d x } \]
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\[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}^{3}}{x \left (d + e x\right )}\, dx \]
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\[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d x} \,d x } \]
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\[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d x} \,d x } \]
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Timed out. \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d x+e x^2} \, dx=\int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^3}{e\,x^2+d\,x} \,d x \]
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