Optimal. Leaf size=210 \[ -\frac{b n \text{PolyLog}(2,-e x)}{4 e^4}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e^3}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}+\frac{1}{4} x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{3 b n x^2}{32 e^2}-\frac{5 b n x}{16 e^3}+\frac{b n \log (e x+1)}{16 e^4}-\frac{7 b n x^3}{144 e}-\frac{1}{16} b n x^4 \log (e x+1)+\frac{1}{32} b n x^4 \]
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Rubi [A] time = 0.119399, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2395, 43, 2376, 2391} \[ -\frac{b n \text{PolyLog}(2,-e x)}{4 e^4}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e^3}-\frac{\log (e x+1) \left (a+b \log \left (c x^n\right )\right )}{4 e^4}+\frac{1}{4} x^4 \log (e x+1) \left (a+b \log \left (c x^n\right )\right )+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{3 b n x^2}{32 e^2}-\frac{5 b n x}{16 e^3}+\frac{b n \log (e x+1)}{16 e^4}-\frac{7 b n x^3}{144 e}-\frac{1}{16} b n x^4 \log (e x+1)+\frac{1}{32} b n x^4 \]
Antiderivative was successfully verified.
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Rule 2395
Rule 43
Rule 2376
Rule 2391
Rubi steps
\begin{align*} \int x^3 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x) \, dx &=\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-(b n) \int \left (\frac{1}{4 e^3}-\frac{x}{8 e^2}+\frac{x^2}{12 e}-\frac{x^3}{16}-\frac{\log (1+e x)}{4 e^4 x}+\frac{1}{4} x^3 \log (1+e x)\right ) \, dx\\ &=-\frac{b n x}{4 e^3}+\frac{b n x^2}{16 e^2}-\frac{b n x^3}{36 e}+\frac{1}{64} b n x^4+\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{1}{4} (b n) \int x^3 \log (1+e x) \, dx+\frac{(b n) \int \frac{\log (1+e x)}{x} \, dx}{4 e^4}\\ &=-\frac{b n x}{4 e^3}+\frac{b n x^2}{16 e^2}-\frac{b n x^3}{36 e}+\frac{1}{64} b n x^4+\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b n x^4 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{b n \text{Li}_2(-e x)}{4 e^4}+\frac{1}{16} (b e n) \int \frac{x^4}{1+e x} \, dx\\ &=-\frac{b n x}{4 e^3}+\frac{b n x^2}{16 e^2}-\frac{b n x^3}{36 e}+\frac{1}{64} b n x^4+\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )-\frac{1}{16} b n x^4 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{b n \text{Li}_2(-e x)}{4 e^4}+\frac{1}{16} (b e n) \int \left (-\frac{1}{e^4}+\frac{x}{e^3}-\frac{x^2}{e^2}+\frac{x^3}{e}+\frac{1}{e^4 (1+e x)}\right ) \, dx\\ &=-\frac{5 b n x}{16 e^3}+\frac{3 b n x^2}{32 e^2}-\frac{7 b n x^3}{144 e}+\frac{1}{32} b n x^4+\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 e^3}-\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{8 e^2}+\frac{x^3 \left (a+b \log \left (c x^n\right )\right )}{12 e}-\frac{1}{16} x^4 \left (a+b \log \left (c x^n\right )\right )+\frac{b n \log (1+e x)}{16 e^4}-\frac{1}{16} b n x^4 \log (1+e x)-\frac{\left (a+b \log \left (c x^n\right )\right ) \log (1+e x)}{4 e^4}+\frac{1}{4} x^4 \left (a+b \log \left (c x^n\right )\right ) \log (1+e x)-\frac{b n \text{Li}_2(-e x)}{4 e^4}\\ \end{align*}
Mathematica [A] time = 0.0899821, size = 188, normalized size = 0.9 \[ \frac{-72 b n \text{PolyLog}(2,-e x)-18 a e^4 x^4+24 a e^3 x^3-36 a e^2 x^2+72 a e^4 x^4 \log (e x+1)+72 a e x-72 a \log (e x+1)+6 b \left (e x \left (-3 e^3 x^3+4 e^2 x^2-6 e x+12\right )+12 \left (e^4 x^4-1\right ) \log (e x+1)\right ) \log \left (c x^n\right )+9 b e^4 n x^4-14 b e^3 n x^3+27 b e^2 n x^2-18 b e^4 n x^4 \log (e x+1)-90 b e n x+18 b n \log (e x+1)}{288 e^4} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.102, size = 1014, normalized size = 4.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.36487, size = 350, normalized size = 1.67 \begin{align*} -\frac{{\left (\log \left (e x + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-e x\right )\right )} b n}{4 \, e^{4}} + \frac{{\left (b{\left (n - 4 \, \log \left (c\right )\right )} - 4 \, a\right )} \log \left (e x + 1\right )}{16 \, e^{4}} - \frac{9 \,{\left (2 \, a e^{4} -{\left (e^{4} n - 2 \, e^{4} \log \left (c\right )\right )} b\right )} x^{4} - 2 \,{\left (12 \, a e^{3} -{\left (7 \, e^{3} n - 12 \, e^{3} \log \left (c\right )\right )} b\right )} x^{3} + 9 \,{\left (4 \, a e^{2} -{\left (3 \, e^{2} n - 4 \, e^{2} \log \left (c\right )\right )} b\right )} x^{2} + 18 \,{\left ({\left (5 \, e n - 4 \, e \log \left (c\right )\right )} b - 4 \, a e\right )} x - 18 \,{\left ({\left (4 \, a e^{4} -{\left (e^{4} n - 4 \, e^{4} \log \left (c\right )\right )} b\right )} x^{4} + 4 \, b n \log \left (x\right )\right )} \log \left (e x + 1\right ) + 6 \,{\left (3 \, b e^{4} x^{4} - 4 \, b e^{3} x^{3} + 6 \, b e^{2} x^{2} - 12 \, b e x - 12 \,{\left (b e^{4} x^{4} - b\right )} \log \left (e x + 1\right )\right )} \log \left (x^{n}\right )}{288 \, e^{4}} \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 (b x^{3} \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a x^{3} \log \left (e x + 1\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3} \log \left (e x + 1\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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