Optimal. Leaf size=81 \[ -6 b^2 n^2 \text {Li}_4(-e x) \left (a+b \log \left (c x^n\right )\right )+3 b n \text {Li}_3(-e x) \left (a+b \log \left (c x^n\right )\right )^2-\text {Li}_2(-e x) \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^3 \text {Li}_5(-e x) \]
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Rubi [A] time = 0.10, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {2374, 2383, 6589} \[ -6 b^2 n^2 \text {PolyLog}(4,-e x) \left (a+b \log \left (c x^n\right )\right )+3 b n \text {PolyLog}(3,-e x) \left (a+b \log \left (c x^n\right )\right )^2-\text {PolyLog}(2,-e x) \left (a+b \log \left (c x^n\right )\right )^3+6 b^3 n^3 \text {PolyLog}(5,-e x) \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log (1+e x)}{x} \, dx &=-\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2(-e x)+(3 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2(-e x)}{x} \, dx\\ &=-\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2(-e x)+3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3(-e x)-\left (6 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3(-e x)}{x} \, dx\\ &=-\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2(-e x)+3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3(-e x)-6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(-e x)+\left (6 b^3 n^3\right ) \int \frac {\text {Li}_4(-e x)}{x} \, dx\\ &=-\left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2(-e x)+3 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3(-e x)-6 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4(-e x)+6 b^3 n^3 \text {Li}_5(-e x)\\ \end {align*}
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Mathematica [A] time = 0.12, size = 77, normalized size = 0.95 \[ 3 b n \left (\text {Li}_3(-e x) \left (a+b \log \left (c x^n\right )\right )^2+2 b n \left (b n \text {Li}_5(-e x)-\text {Li}_4(-e x) \left (a+b \log \left (c x^n\right )\right )\right )\right )-\text {Li}_2(-e x) \left (a+b \log \left (c x^n\right )\right )^3 \]
Antiderivative was successfully verified.
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fricas [F] time = 0.78, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{3} \log \left (c x^{n}\right )^{3} \log \left (e x + 1\right ) + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} \log \left (e x + 1\right ) + 3 \, a^{2} b \log \left (c x^{n}\right ) \log \left (e x + 1\right ) + a^{3} \log \left (e x + 1\right )}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (e x +1\right )}{x}\, dx \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left (e x + 1\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (e\,x+1\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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