Optimal. Leaf size=161 \[ \frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {6 b n p^2 \log \left (f x^p\right ) \text {Li}_4\left (-\frac {e x^m}{d}\right )}{m^3}+\frac {3 b n p \log ^2\left (f x^p\right ) \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}-\frac {b n \log ^3\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}-\frac {b n \log ^4\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{4 p}+\frac {6 b n p^3 \text {Li}_5\left (-\frac {e x^m}{d}\right )}{m^4} \]
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Rubi [A] time = 0.22, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2481, 2337, 2374, 2383, 6589} \[ -\frac {6 b n p^2 \log \left (f x^p\right ) \text {PolyLog}\left (4,-\frac {e x^m}{d}\right )}{m^3}+\frac {3 b n p \log ^2\left (f x^p\right ) \text {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}-\frac {b n \log ^3\left (f x^p\right ) \text {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}+\frac {6 b n p^3 \text {PolyLog}\left (5,-\frac {e x^m}{d}\right )}{m^4}+\frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b n \log ^4\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{4 p} \]
Antiderivative was successfully verified.
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Rule 2337
Rule 2374
Rule 2383
Rule 2481
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx &=\frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {(b e m n) \int \frac {x^{-1+m} \log ^4\left (f x^p\right )}{d+e x^m} \, dx}{4 p}\\ &=\frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b n \log ^4\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{4 p}+(b n) \int \frac {\log ^3\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{x} \, dx\\ &=\frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b n \log ^4\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{4 p}-\frac {b n \log ^3\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {(3 b n p) \int \frac {\log ^2\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{x} \, dx}{m}\\ &=\frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b n \log ^4\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{4 p}-\frac {b n \log ^3\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {3 b n p \log ^2\left (f x^p\right ) \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}-\frac {\left (6 b n p^2\right ) \int \frac {\log \left (f x^p\right ) \text {Li}_3\left (-\frac {e x^m}{d}\right )}{x} \, dx}{m^2}\\ &=\frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b n \log ^4\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{4 p}-\frac {b n \log ^3\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {3 b n p \log ^2\left (f x^p\right ) \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}-\frac {6 b n p^2 \log \left (f x^p\right ) \text {Li}_4\left (-\frac {e x^m}{d}\right )}{m^3}+\frac {\left (6 b n p^3\right ) \int \frac {\text {Li}_4\left (-\frac {e x^m}{d}\right )}{x} \, dx}{m^3}\\ &=\frac {\log ^4\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{4 p}-\frac {b n \log ^4\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{4 p}-\frac {b n \log ^3\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {3 b n p \log ^2\left (f x^p\right ) \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}-\frac {6 b n p^2 \log \left (f x^p\right ) \text {Li}_4\left (-\frac {e x^m}{d}\right )}{m^3}+\frac {6 b n p^3 \text {Li}_5\left (-\frac {e x^m}{d}\right )}{m^4}\\ \end {align*}
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Mathematica [B] time = 0.28, size = 659, normalized size = 4.09 \[ \frac {a \log ^4\left (f x^p\right )}{4 p}+b p^2 \log ^3(x) \log \left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )+b \log (x) \log ^3\left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )-\frac {3}{2} b p \log ^2(x) \log ^2\left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )-\frac {1}{4} b p^3 \log ^4(x) \log \left (c \left (d+e x^m\right )^n\right )+\frac {6 b n p^2 \log \left (f x^p\right ) \text {Li}_4\left (-\frac {d x^{-m}}{e}\right )}{m^3}+\frac {3 b n p \log ^2\left (f x^p\right ) \text {Li}_3\left (-\frac {d x^{-m}}{e}\right )}{m^2}+\frac {b n p \log (x) \left (3 \log ^2\left (f x^p\right )-3 p \log (x) \log \left (f x^p\right )+p^2 \log ^2(x)\right ) \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )}{m}+2 b n p^2 \log ^3(x) \log \left (f x^p\right ) \log \left (\frac {d x^{-m}}{e}+1\right )-3 b n p^2 \log ^3(x) \log \left (f x^p\right ) \log \left (d+e x^m\right )+\frac {3 b n p^2 \log ^2(x) \log \left (f x^p\right ) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}-\frac {b n \left (p \log (x)-\log \left (f x^p\right )\right )^3 \text {Li}_2\left (\frac {e x^m}{d}+1\right )}{m}-\frac {3}{2} b n p \log ^2(x) \log ^2\left (f x^p\right ) \log \left (\frac {d x^{-m}}{e}+1\right )-b n \log (x) \log ^3\left (f x^p\right ) \log \left (d+e x^m\right )+\frac {b n \log ^3\left (f x^p\right ) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}+3 b n p \log ^2(x) \log ^2\left (f x^p\right ) \log \left (d+e x^m\right )-\frac {3 b n p \log (x) \log ^2\left (f x^p\right ) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}+\frac {6 b n p^3 \text {Li}_5\left (-\frac {d x^{-m}}{e}\right )}{m^4}-\frac {3}{4} b n p^3 \log ^4(x) \log \left (\frac {d x^{-m}}{e}+1\right )+b n p^3 \log ^4(x) \log \left (d+e x^m\right )-\frac {b n p^3 \log ^3(x) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}+\frac {3}{4} b m n p^2 \log ^4(x) \log \left (f x^p\right )-\frac {1}{2} b m n p \log ^3(x) \log ^2\left (f x^p\right )-\frac {3}{10} b m n p^3 \log ^5(x) \]
Antiderivative was successfully verified.
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fricas [C] time = 1.29, size = 417, normalized size = 2.59 \[ \frac {24 \, b n p^{3} {\rm polylog}\left (5, -\frac {e x^{m}}{d}\right ) + 4 \, {\left (b m^{4} \log \relax (c) + a m^{4}\right )} \log \relax (f)^{3} \log \relax (x) + 6 \, {\left (b m^{4} p \log \relax (c) + a m^{4} p\right )} \log \relax (f)^{2} \log \relax (x)^{2} + 4 \, {\left (b m^{4} p^{2} \log \relax (c) + a m^{4} p^{2}\right )} \log \relax (f) \log \relax (x)^{3} + {\left (b m^{4} p^{3} \log \relax (c) + a m^{4} p^{3}\right )} \log \relax (x)^{4} - 4 \, {\left (b m^{3} n p^{3} \log \relax (x)^{3} + 3 \, b m^{3} n p^{2} \log \relax (f) \log \relax (x)^{2} + 3 \, b m^{3} n p \log \relax (f)^{2} \log \relax (x) + b m^{3} n \log \relax (f)^{3}\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) + {\left (b m^{4} n p^{3} \log \relax (x)^{4} + 4 \, b m^{4} n p^{2} \log \relax (f) \log \relax (x)^{3} + 6 \, b m^{4} n p \log \relax (f)^{2} \log \relax (x)^{2} + 4 \, b m^{4} n \log \relax (f)^{3} \log \relax (x)\right )} \log \left (e x^{m} + d\right ) - {\left (b m^{4} n p^{3} \log \relax (x)^{4} + 4 \, b m^{4} n p^{2} \log \relax (f) \log \relax (x)^{3} + 6 \, b m^{4} n p \log \relax (f)^{2} \log \relax (x)^{2} + 4 \, b m^{4} n \log \relax (f)^{3} \log \relax (x)\right )} \log \left (\frac {e x^{m} + d}{d}\right ) - 24 \, {\left (b m n p^{3} \log \relax (x) + b m n p^{2} \log \relax (f)\right )} {\rm polylog}\left (4, -\frac {e x^{m}}{d}\right ) + 12 \, {\left (b m^{2} n p^{3} \log \relax (x)^{2} + 2 \, b m^{2} n p^{2} \log \relax (f) \log \relax (x) + b m^{2} n p \log \relax (f)^{2}\right )} {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right )}{4 \, m^{4}} \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{p}\right )^{3}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.20, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e \,x^{m}+d \right )^{n}\right )+a \right ) \ln \left (f \,x^{p}\right )^{3}}{x}\, dx \]
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 \[ -\frac {1}{4} \, {\left (b p^{3} \log \relax (x)^{4} - 4 \, b p^{2} \log \relax (f) \log \relax (x)^{3} + 6 \, b p \log \relax (f)^{2} \log \relax (x)^{2} - 4 \, b \log \relax (f)^{3} \log \relax (x) - 4 \, b \log \relax (x) \log \left (x^{p}\right )^{3} + 6 \, {\left (b p \log \relax (x)^{2} - 2 \, b \log \relax (f) \log \relax (x)\right )} \log \left (x^{p}\right )^{2} - 4 \, {\left (b p^{2} \log \relax (x)^{3} - 3 \, b p \log \relax (f) \log \relax (x)^{2} + 3 \, b \log \relax (f)^{2} \log \relax (x)\right )} \log \left (x^{p}\right )\right )} \log \left ({\left (e x^{m} + d\right )}^{n}\right ) - \int -\frac {4 \, b d \log \relax (c) \log \relax (f)^{3} + 4 \, a d \log \relax (f)^{3} + 4 \, {\left (b d \log \relax (c) + a d - {\left (b e m n \log \relax (x) - b e \log \relax (c) - a e\right )} x^{m}\right )} \log \left (x^{p}\right )^{3} + 6 \, {\left (2 \, b d \log \relax (c) \log \relax (f) + 2 \, a d \log \relax (f) + {\left (b e m n p \log \relax (x)^{2} - 2 \, b e m n \log \relax (f) \log \relax (x) + 2 \, b e \log \relax (c) \log \relax (f) + 2 \, a e \log \relax (f)\right )} x^{m}\right )} \log \left (x^{p}\right )^{2} + {\left (b e m n p^{3} \log \relax (x)^{4} - 4 \, b e m n p^{2} \log \relax (f) \log \relax (x)^{3} + 6 \, b e m n p \log \relax (f)^{2} \log \relax (x)^{2} - 4 \, b e m n \log \relax (f)^{3} \log \relax (x) + 4 \, b e \log \relax (c) \log \relax (f)^{3} + 4 \, a e \log \relax (f)^{3}\right )} x^{m} + 4 \, {\left (3 \, b d \log \relax (c) \log \relax (f)^{2} + 3 \, a d \log \relax (f)^{2} - {\left (b e m n p^{2} \log \relax (x)^{3} - 3 \, b e m n p \log \relax (f) \log \relax (x)^{2} + 3 \, b e m n \log \relax (f)^{2} \log \relax (x) - 3 \, b e \log \relax (c) \log \relax (f)^{2} - 3 \, a e \log \relax (f)^{2}\right )} x^{m}\right )} \log \left (x^{p}\right )}{4 \, {\left (e x x^{m} + d x\right )}}\,{d x} \]
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 \[ \int \frac {{\ln \left (f\,x^p\right )}^3\,\left (a+b\,\ln \left (c\,{\left (d+e\,x^m\right )}^n\right )\right )}{x} \,d x \]
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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