Optimal. Leaf size=477 \[ \frac{3 n^2 \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (3,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{3 n^2 \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (3,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{3 n^3 \text{PolyLog}\left (4,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{3 n^3 \text{PolyLog}\left (4,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{\log ^3\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}-\sqrt{e} x\right )}{a \sqrt{e}+b \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log ^3\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}+\sqrt{e} x\right )}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}} \]
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Rubi [A] time = 0.537332, antiderivative size = 477, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2409, 2396, 2433, 2374, 2383, 6589} \[ \frac{3 n^2 \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (3,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{3 n^2 \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (3,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}-\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{PolyLog}\left (2,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}}+\frac{3 n \log ^2\left (c (a+b x)^n\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{3 n^3 \text{PolyLog}\left (4,-\frac{\sqrt{e} (a+b x)}{b \sqrt{-d}-a \sqrt{e}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{3 n^3 \text{PolyLog}\left (4,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+b \sqrt{-d}}\right )}{\sqrt{-d} \sqrt{e}}+\frac{\log ^3\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}-\sqrt{e} x\right )}{a \sqrt{e}+b \sqrt{-d}}\right )}{2 \sqrt{-d} \sqrt{e}}-\frac{\log ^3\left (c (a+b x)^n\right ) \log \left (\frac{b \left (\sqrt{-d}+\sqrt{e} x\right )}{b \sqrt{-d}-a \sqrt{e}}\right )}{2 \sqrt{-d} \sqrt{e}} \]
Antiderivative was successfully verified.
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Rule 2409
Rule 2396
Rule 2433
Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \frac{\log ^3\left (c (a+b x)^n\right )}{d+e x^2} \, dx &=\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) \operatorname{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) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}
Mathematica [C] time = 0.2752, size = 754, normalized size = 1.58 \[ \frac{-6 i n^2 \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (3,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}-i b \sqrt{d}}\right )+6 i n^2 \log \left (c (a+b x)^n\right ) \text{PolyLog}\left (3,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+i b \sqrt{d}}\right )+3 i n \log ^2\left (c (a+b x)^n\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}-i b \sqrt{d}}\right )-3 i n \log ^2\left (c (a+b x)^n\right ) \text{PolyLog}\left (2,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+i b \sqrt{d}}\right )+6 i n^3 \text{PolyLog}\left (4,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}-i b \sqrt{d}}\right )-6 i n^3 \text{PolyLog}\left (4,\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+i b \sqrt{d}}\right )-3 i n^2 \log ^2(a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac{\sqrt{e} (a+b x)}{a \sqrt{e}-i b \sqrt{d}}\right )+3 i n^2 \log ^2(a+b x) \log \left (c (a+b x)^n\right ) \log \left (1-\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+i b \sqrt{d}}\right )+6 n^2 \log ^2(a+b x) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log \left (c (a+b x)^n\right )+3 i n \log (a+b x) \log ^2\left (c (a+b x)^n\right ) \log \left (1-\frac{\sqrt{e} (a+b x)}{a \sqrt{e}-i b \sqrt{d}}\right )-3 i n \log (a+b x) \log ^2\left (c (a+b x)^n\right ) \log \left (1-\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+i b \sqrt{d}}\right )-6 n \log (a+b x) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log ^2\left (c (a+b x)^n\right )+2 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \log ^3\left (c (a+b x)^n\right )+i n^3 \log ^3(a+b x) \log \left (1-\frac{\sqrt{e} (a+b x)}{a \sqrt{e}-i b \sqrt{d}}\right )-i n^3 \log ^3(a+b x) \log \left (1-\frac{\sqrt{e} (a+b x)}{a \sqrt{e}+i b \sqrt{d}}\right )-2 n^3 \log ^3(a+b x) \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 \sqrt{d} \sqrt{e}} \]
Antiderivative was successfully verified.
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Maple [F] time = 4.885, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( bx+a \right ) ^{n} \right ) \right ) ^{3}}{e{x}^{2}+d}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (c \left (a + b x\right )^{n} \right )}^{3}}{d + e x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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