Optimal. Leaf size=132 \[ \frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}+\frac {2 b n p \log \left (f x^p\right ) \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}-\frac {b n \log ^2\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}-\frac {b n \log ^3\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{3 p}-\frac {2 b n p^2 \text {Li}_4\left (-\frac {e x^m}{d}\right )}{m^3} \]
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Rubi [A] time = 0.19, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ \frac {2 b n p \log \left (f x^p\right ) \text {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}-\frac {b n \log ^2\left (f x^p\right ) \text {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}-\frac {2 b n p^2 \text {PolyLog}\left (4,-\frac {e x^m}{d}\right )}{m^3}+\frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {b n \log ^3\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{3 p} \]
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 ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx &=\frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {(b e m n) \int \frac {x^{-1+m} \log ^3\left (f x^p\right )}{d+e x^m} \, dx}{3 p}\\ &=\frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {b n \log ^3\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{3 p}+(b n) \int \frac {\log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{x} \, dx\\ &=\frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {b n \log ^3\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{3 p}-\frac {b n \log ^2\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {(2 b n p) \int \frac {\log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{x} \, dx}{m}\\ &=\frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {b n \log ^3\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{3 p}-\frac {b n \log ^2\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {2 b n p \log \left (f x^p\right ) \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}-\frac {\left (2 b n p^2\right ) \int \frac {\text {Li}_3\left (-\frac {e x^m}{d}\right )}{x} \, dx}{m^2}\\ &=\frac {\log ^3\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{3 p}-\frac {b n \log ^3\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{3 p}-\frac {b n \log ^2\left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {2 b n p \log \left (f x^p\right ) \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}-\frac {2 b n p^2 \text {Li}_4\left (-\frac {e x^m}{d}\right )}{m^3}\\ \end {align*}
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Mathematica [B] time = 0.25, size = 456, normalized size = 3.45 \[ \frac {a \log ^3\left (f x^p\right )}{3 p}-b p \log ^2(x) \log \left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )+b \log (x) \log ^2\left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )+\frac {1}{3} b p^2 \log ^3(x) \log \left (c \left (d+e x^m\right )^n\right )+\frac {2 b n p \log \left (f x^p\right ) \text {Li}_3\left (-\frac {d x^{-m}}{e}\right )}{m^2}-\frac {b n p \log (x) \left (p \log (x)-2 \log \left (f x^p\right )\right ) \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )}{m}+\frac {b n \left (\log \left (f x^p\right )-p \log (x)\right )^2 \text {Li}_2\left (\frac {e x^m}{d}+1\right )}{m}-b n p \log ^2(x) \log \left (f x^p\right ) \log \left (\frac {d x^{-m}}{e}+1\right )+2 b n p \log ^2(x) \log \left (f x^p\right ) \log \left (d+e x^m\right )-b n \log (x) \log ^2\left (f x^p\right ) \log \left (d+e x^m\right )+\frac {b n \log ^2\left (f x^p\right ) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}-\frac {2 b n p \log (x) \log \left (f x^p\right ) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}+\frac {2 b n p^2 \text {Li}_4\left (-\frac {d x^{-m}}{e}\right )}{m^3}+\frac {2}{3} b n p^2 \log ^3(x) \log \left (\frac {d x^{-m}}{e}+1\right )-b n p^2 \log ^3(x) \log \left (d+e x^m\right )+\frac {b n p^2 \log ^2(x) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}-\frac {1}{3} b m n p \log ^3(x) \log \left (f x^p\right )+\frac {1}{4} b m n p^2 \log ^4(x) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.72, size = 281, normalized size = 2.13 \[ -\frac {6 \, b n p^{2} {\rm polylog}\left (4, -\frac {e x^{m}}{d}\right ) - 3 \, {\left (b m^{3} \log \relax (c) + a m^{3}\right )} \log \relax (f)^{2} \log \relax (x) - 3 \, {\left (b m^{3} p \log \relax (c) + a m^{3} p\right )} \log \relax (f) \log \relax (x)^{2} - {\left (b m^{3} p^{2} \log \relax (c) + a m^{3} p^{2}\right )} \log \relax (x)^{3} + 3 \, {\left (b m^{2} n p^{2} \log \relax (x)^{2} + 2 \, b m^{2} n p \log \relax (f) \log \relax (x) + b m^{2} n \log \relax (f)^{2}\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) - {\left (b m^{3} n p^{2} \log \relax (x)^{3} + 3 \, b m^{3} n p \log \relax (f) \log \relax (x)^{2} + 3 \, b m^{3} n \log \relax (f)^{2} \log \relax (x)\right )} \log \left (e x^{m} + d\right ) + {\left (b m^{3} n p^{2} \log \relax (x)^{3} + 3 \, b m^{3} n p \log \relax (f) \log \relax (x)^{2} + 3 \, b m^{3} n \log \relax (f)^{2} \log \relax (x)\right )} \log \left (\frac {e x^{m} + d}{d}\right ) - 6 \, {\left (b m n p^{2} \log \relax (x) + b m n p \log \relax (f)\right )} {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right )}{3 \, m^{3}} \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{p}\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \left (e \,x^{m}+d \right )^{n}\right )+a \right ) \ln \left (f \,x^{p}\right )^{2}}{x}\, dx \]
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 \[ \frac {1}{3} \, {\left (b p^{2} \log \relax (x)^{3} - 3 \, b p \log \relax (f) \log \relax (x)^{2} + 3 \, b \log \relax (f)^{2} \log \relax (x) + 3 \, b \log \relax (x) \log \left (x^{p}\right )^{2} - 3 \, {\left (b p \log \relax (x)^{2} - 2 \, b \log \relax (f) \log \relax (x)\right )} \log \left (x^{p}\right )\right )} \log \left ({\left (e x^{m} + d\right )}^{n}\right ) - \int -\frac {3 \, b d \log \relax (c) \log \relax (f)^{2} + 3 \, a d \log \relax (f)^{2} + 3 \, {\left (b d \log \relax (c) + a d - {\left (b e m n \log \relax (x) - b e \log \relax (c) - a e\right )} x^{m}\right )} \log \left (x^{p}\right )^{2} - {\left (b e m n p^{2} \log \relax (x)^{3} - 3 \, b e m n p \log \relax (f) \log \relax (x)^{2} + 3 \, b e m n \log \relax (f)^{2} \log \relax (x) - 3 \, b e \log \relax (c) \log \relax (f)^{2} - 3 \, a e \log \relax (f)^{2}\right )} x^{m} + 3 \, {\left (2 \, b d \log \relax (c) \log \relax (f) + 2 \, a d \log \relax (f) + {\left (b e m n p \log \relax (x)^{2} - 2 \, b e m n \log \relax (f) \log \relax (x) + 2 \, b e \log \relax (c) \log \relax (f) + 2 \, a e \log \relax (f)\right )} x^{m}\right )} \log \left (x^{p}\right )}{3 \, {\left (e x x^{m} + d x\right )}}\,{d x} \]
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 \[ \int \frac {{\ln \left (f\,x^p\right )}^2\,\left (a+b\,\ln \left (c\,{\left (d+e\,x^m\right )}^n\right )\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \log {\left (c \left (d + e x^{m}\right )^{n} \right )}\right ) \log {\left (f x^{p} \right )}^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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