Optimal. Leaf size=161 \[ -6 b^2 m n^2 \text {Li}_4\left (-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-m \text {Li}_2\left (-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+3 b m n \text {Li}_3\left (-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}+6 b^3 m n^3 \text {Li}_5\left (-\frac {f x}{e}\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {2375, 2317, 2374, 2383, 6589} \[ -6 b^2 m n^2 \text {PolyLog}\left (4,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-m \text {PolyLog}\left (2,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+3 b m n \text {PolyLog}\left (3,-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+6 b^3 m n^3 \text {PolyLog}\left (5,-\frac {f x}{e}\right )+\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \log \left (\frac {f x}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]
Antiderivative was successfully verified.
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Rule 2317
Rule 2374
Rule 2375
Rule 2383
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (d (e+f x)^m\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {(f m) \int \frac {\left (a+b \log \left (c x^n\right )\right )^4}{e+f x} \, dx}{4 b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}+m \int \frac {\left (a+b \log \left (c x^n\right )\right )^3 \log \left (1+\frac {f x}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f x}{e}\right )+(3 b m n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f x}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f x}{e}\right )-\left (6 b^2 m n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f x}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f x}{e}\right )-6 b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-\frac {f x}{e}\right )+\left (6 b^3 m n^3\right ) \int \frac {\text {Li}_4\left (-\frac {f x}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^4 \log \left (d (e+f x)^m\right )}{4 b n}-\frac {m \left (a+b \log \left (c x^n\right )\right )^4 \log \left (1+\frac {f x}{e}\right )}{4 b n}-m \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f x}{e}\right )+3 b m n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f x}{e}\right )-6 b^2 m n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-\frac {f x}{e}\right )+6 b^3 m n^3 \text {Li}_5\left (-\frac {f x}{e}\right )\\ \end {align*}
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Mathematica [B] time = 0.26, size = 602, normalized size = 3.74 \[ a^3 \log (x) \log \left (d (e+f x)^m\right )-a^3 m \log (x) \log \left (\frac {f x}{e}+1\right )+3 a^2 b \log (x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )-3 a^2 b m \log (x) \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )-\frac {3}{2} a^2 b n \log ^2(x) \log \left (d (e+f x)^m\right )+\frac {3}{2} a^2 b m n \log ^2(x) \log \left (\frac {f x}{e}+1\right )-3 a b^2 n \log ^2(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+3 a b^2 \log (x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )+3 a b^2 m n \log ^2(x) \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )-3 a b^2 m \log (x) \log ^2\left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )+a b^2 n^2 \log ^3(x) \log \left (d (e+f x)^m\right )-6 a b^2 m n^2 \text {Li}_4\left (-\frac {f x}{e}\right )-a b^2 m n^2 \log ^3(x) \log \left (\frac {f x}{e}+1\right )-m \text {Li}_2\left (-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+3 b m n \text {Li}_3\left (-\frac {f x}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+b^3 n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (d (e+f x)^m\right )+b^3 \log (x) \log ^3\left (c x^n\right ) \log \left (d (e+f x)^m\right )-\frac {3}{2} b^3 n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (d (e+f x)^m\right )-6 b^3 m n^2 \log \left (c x^n\right ) \text {Li}_4\left (-\frac {f x}{e}\right )-b^3 m n^2 \log ^3(x) \log \left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )-b^3 m \log (x) \log ^3\left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )+\frac {3}{2} b^3 m n \log ^2(x) \log ^2\left (c x^n\right ) \log \left (\frac {f x}{e}+1\right )-\frac {1}{4} b^3 n^3 \log ^4(x) \log \left (d (e+f x)^m\right )+6 b^3 m n^3 \text {Li}_5\left (-\frac {f x}{e}\right )+\frac {1}{4} b^3 m n^3 \log ^4(x) \log \left (\frac {f x}{e}+1\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left ({\left (f x + e\right )}^{m} d\right )}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f x + e\right )}^{m} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.96, size = 60520, normalized size = 375.90 \[ \text {output too large to display} \]
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 \[ \text {result too large to display} \]
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 \[ \int \frac {\ln \left (d\,{\left (e+f\,x\right )}^m\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x} \,d x \]
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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