Integrand size = 22, antiderivative size = 477 \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}+\sqrt {e} x\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {3 n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}} \]
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Time = 0.39 (sec) , antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2456, 2443, 2481, 2421, 2430, 6724} \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\frac {3 n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {e} (a+b x)}{\sqrt {e} a+b \sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (a+b x)}{\sqrt {e} a+b \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{a \sqrt {e}+b \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}+\sqrt {e} x\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {3 n^3 \operatorname {PolyLog}\left (4,-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {3 n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} (a+b x)}{\sqrt {e} a+b \sqrt {-d}}\right )}{\sqrt {-d} \sqrt {e}} \]
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Rule 2421
Rule 2430
Rule 2443
Rule 2456
Rule 2481
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\sqrt {-d} \log ^3\left (c (a+b x)^n\right )}{2 d \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} \log ^3\left (c (a+b x)^n\right )}{2 d \left (\sqrt {-d}+\sqrt {e} x\right )}\right ) \, dx \\ & = -\frac {\int \frac {\log ^3\left (c (a+b x)^n\right )}{\sqrt {-d}-\sqrt {e} x} \, dx}{2 \sqrt {-d}}-\frac {\int \frac {\log ^3\left (c (a+b x)^n\right )}{\sqrt {-d}+\sqrt {e} x} \, dx}{2 \sqrt {-d}} \\ & = \frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}+\sqrt {e} x\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(3 b n) \int \frac {\log ^2\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{a+b x} \, dx}{2 \sqrt {-d} \sqrt {e}}+\frac {(3 b n) \int \frac {\log ^2\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}+\sqrt {e} x\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{a+b x} \, dx}{2 \sqrt {-d} \sqrt {e}} \\ & = \frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}+\sqrt {e} x\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {(3 n) \text {Subst}\left (\int \frac {\log ^2\left (c x^n\right ) \log \left (\frac {b \left (\frac {b \sqrt {-d}+a \sqrt {e}}{b}-\frac {\sqrt {e} x}{b}\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{x} \, dx,x,a+b x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {(3 n) \text {Subst}\left (\int \frac {\log ^2\left (c x^n\right ) \log \left (\frac {b \left (\frac {b \sqrt {-d}-a \sqrt {e}}{b}+\frac {\sqrt {e} x}{b}\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{x} \, dx,x,a+b x\right )}{2 \sqrt {-d} \sqrt {e}} \\ & = \frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}+\sqrt {e} x\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^n\right ) \text {Li}_2\left (-\frac {\sqrt {e} x}{b \sqrt {-d}-a \sqrt {e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {-d} \sqrt {e}}-\frac {\left (3 n^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^n\right ) \text {Li}_2\left (\frac {\sqrt {e} x}{b \sqrt {-d}+a \sqrt {e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {-d} \sqrt {e}} \\ & = \frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}+\sqrt {e} x\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {\left (3 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {\sqrt {e} x}{b \sqrt {-d}-a \sqrt {e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {-d} \sqrt {e}}+\frac {\left (3 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {\sqrt {e} x}{b \sqrt {-d}+a \sqrt {e}}\right )}{x} \, dx,x,a+b x\right )}{\sqrt {-d} \sqrt {e}} \\ & = \frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}-\sqrt {e} x\right )}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b \left (\sqrt {-d}+\sqrt {e} x\right )}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {3 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}-\frac {3 n^3 \text {Li}_4\left (-\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}-a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}}+\frac {3 n^3 \text {Li}_4\left (\frac {\sqrt {e} (a+b x)}{b \sqrt {-d}+a \sqrt {e}}\right )}{\sqrt {-d} \sqrt {e}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.39 (sec) , antiderivative size = 754, normalized size of antiderivative = 1.58 \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\frac {-2 n^3 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log ^3(a+b x)+6 n^2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log ^2(a+b x) \log \left (c (a+b x)^n\right )-6 n \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log (a+b x) \log ^2\left (c (a+b x)^n\right )+2 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \log ^3\left (c (a+b x)^n\right )+i n^3 \log ^3(a+b x) \log \left (1-\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )-3 i n^2 \log ^2(a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )+3 i n \log (a+b x) \log ^2\left (c (a+b x)^n\right ) \log \left (1-\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )-i n^3 \log ^3(a+b x) \log \left (1-\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )+3 i n^2 \log ^2(a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )-3 i n \log (a+b x) \log ^2\left (c (a+b x)^n\right ) \log \left (1-\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )+3 i n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )-3 i n \log ^2\left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (2,\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )-6 i n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )+6 i n^2 \log \left (c (a+b x)^n\right ) \operatorname {PolyLog}\left (3,\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )+6 i n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} (a+b x)}{-i b \sqrt {d}+a \sqrt {e}}\right )-6 i n^3 \operatorname {PolyLog}\left (4,\frac {\sqrt {e} (a+b x)}{i b \sqrt {d}+a \sqrt {e}}\right )}{2 \sqrt {d} \sqrt {e}} \]
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\[\int \frac {\ln \left (c \left (b x +a \right )^{n}\right )^{3}}{e \,x^{2}+d}d x\]
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\[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d} \,d x } \]
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\[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}^{3}}{d + e x^{2}}\, dx \]
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Exception generated. \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\text {Exception raised: ValueError} \]
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\[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\int { \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d} \,d x } \]
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Timed out. \[ \int \frac {\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx=\int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^3}{e\,x^2+d} \,d x \]
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