Optimal. Leaf size=238 \[ \frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {Li}_3\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {Li}_2\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}-\frac {6 n^2 \text {Li}_3\left (\frac {b x}{a}+1\right ) \log \left (c (a+b x)^n\right )}{d}+\frac {3 n \text {Li}_2\left (\frac {b x}{a}+1\right ) \log ^2\left (c (a+b x)^n\right )}{d}+\frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d}-\frac {6 n^3 \text {Li}_4\left (-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {6 n^3 \text {Li}_4\left (\frac {b x}{a}+1\right )}{d} \]
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Rubi [A] time = 0.35, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1593, 2416, 2396, 2433, 2374, 2383, 6589} \[ \frac {6 n^2 \log \left (c (a+b x)^n\right ) \text {PolyLog}\left (3,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {3 n \log ^2\left (c (a+b x)^n\right ) \text {PolyLog}\left (2,-\frac {e (a+b x)}{b d-a e}\right )}{d}-\frac {6 n^2 \text {PolyLog}\left (3,\frac {b x}{a}+1\right ) \log \left (c (a+b x)^n\right )}{d}+\frac {3 n \text {PolyLog}\left (2,\frac {b x}{a}+1\right ) \log ^2\left (c (a+b x)^n\right )}{d}-\frac {6 n^3 \text {PolyLog}\left (4,-\frac {e (a+b x)}{b d-a e}\right )}{d}+\frac {6 n^3 \text {PolyLog}\left (4,\frac {b x}{a}+1\right )}{d}-\frac {\log ^3\left (c (a+b x)^n\right ) \log \left (\frac {b (d+e x)}{b d-a e}\right )}{d}+\frac {\log \left (-\frac {b x}{a}\right ) \log ^3\left (c (a+b x)^n\right )}{d} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 2374
Rule 2383
Rule 2396
Rule 2416
Rule 2433
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log ^3\left (c (a+b x)^n\right )}{d x+e x^2} \, dx &=\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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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*}
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Mathematica [B] time = 0.22, size = 494, normalized size = 2.08 \[ \frac {-3 n^2 \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right ) \left (2 \text {Li}_3\left (\frac {e (a+b x)}{a e-b d}\right )-2 \log (a+b x) \text {Li}_2\left (\frac {e (a+b x)}{a e-b d}\right )-\log ^2(a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-2 \text {Li}_3\left (\frac {b x}{a}+1\right )+2 \text {Li}_2\left (\frac {b x}{a}+1\right ) \log (a+b x)+\log \left (-\frac {b x}{a}\right ) \log ^2(a+b x)\right )+3 n \left (\log \left (c (a+b x)^n\right )-n \log (a+b x)\right )^2 \left (-\text {Li}_2\left (\frac {e (a+b x)}{a e-b d}\right )-\log (a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )-\text {Li}_2\left (-\frac {b x}{a}\right )+\log (x) \left (\log (a+b x)-\log \left (\frac {b x}{a}+1\right )\right )\right )+\log (d+e x) \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right )^3-\log (x) \left (n \log (a+b x)-\log \left (c (a+b x)^n\right )\right )^3+n^3 \left (-6 \text {Li}_4\left (\frac {e (a+b x)}{a e-b d}\right )-3 \log ^2(a+b x) \text {Li}_2\left (\frac {e (a+b x)}{a e-b d}\right )+6 \log (a+b x) \text {Li}_3\left (\frac {e (a+b x)}{a e-b d}\right )-\log ^3(a+b x) \log \left (\frac {b (d+e x)}{b d-a e}\right )+6 \text {Li}_4\left (\frac {b x}{a}+1\right )+3 \text {Li}_2\left (\frac {b x}{a}+1\right ) \log ^2(a+b x)-6 \text {Li}_3\left (\frac {b x}{a}+1\right ) \log (a+b x)+\log \left (-\frac {b x}{a}\right ) \log ^3(a+b x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.08, size = 12205, normalized size = 51.28 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left ({\left (b x + a\right )}^{n} c\right )^{3}}{e x^{2} + d x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (c\,{\left (a+b\,x\right )}^n\right )}^3}{e\,x^2+d\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (c \left (a + b x\right )^{n} \right )}^{3}}{x \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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