Optimal. Leaf size=102 \[ \frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {b n \log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}-\frac {b n \log ^2\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{2 p}+\frac {b n p \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2} \]
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Rubi [A] time = 0.14, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2481, 2337, 2374, 6589} \[ -\frac {b n \log \left (f x^p\right ) \text {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}+\frac {b n p \text {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}+\frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {b n \log ^2\left (f x^p\right ) \log \left (\frac {e x^m}{d}+1\right )}{2 p} \]
Antiderivative was successfully verified.
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Rule 2337
Rule 2374
Rule 2481
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log \left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{x} \, dx &=\frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {(b e m n) \int \frac {x^{-1+m} \log ^2\left (f x^p\right )}{d+e x^m} \, dx}{2 p}\\ &=\frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {b n \log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{2 p}+(b n) \int \frac {\log \left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{x} \, dx\\ &=\frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {b n \log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{2 p}-\frac {b n \log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {(b n p) \int \frac {\text {Li}_2\left (-\frac {e x^m}{d}\right )}{x} \, dx}{m}\\ &=\frac {\log ^2\left (f x^p\right ) \left (a+b \log \left (c \left (d+e x^m\right )^n\right )\right )}{2 p}-\frac {b n \log ^2\left (f x^p\right ) \log \left (1+\frac {e x^m}{d}\right )}{2 p}-\frac {b n \log \left (f x^p\right ) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {b n p \text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}\\ \end {align*}
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Mathematica [B] time = 0.19, size = 265, normalized size = 2.60 \[ \frac {a \log ^2\left (f x^p\right )}{2 p}+b \log (x) \log \left (f x^p\right ) \log \left (c \left (d+e x^m\right )^n\right )-\frac {1}{2} b p \log ^2(x) \log \left (c \left (d+e x^m\right )^n\right )-\frac {b n \left (p \log (x)-\log \left (f x^p\right )\right ) \text {Li}_2\left (\frac {e x^m}{d}+1\right )}{m}-b n \log (x) \log \left (f x^p\right ) \log \left (d+e x^m\right )+\frac {b n \log \left (f x^p\right ) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}+\frac {b n p \text {Li}_3\left (-\frac {d x^{-m}}{e}\right )}{m^2}+\frac {b n p \log (x) \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )}{m}-\frac {1}{2} b n p \log ^2(x) \log \left (\frac {d x^{-m}}{e}+1\right )+b n p \log ^2(x) \log \left (d+e x^m\right )-\frac {b n p \log (x) \log \left (-\frac {e x^m}{d}\right ) \log \left (d+e x^m\right )}{m}-\frac {1}{6} b m n p \log ^3(x) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.74, size = 161, normalized size = 1.58 \[ \frac {2 \, b n p {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right ) + 2 \, {\left (b m^{2} \log \relax (c) + a m^{2}\right )} \log \relax (f) \log \relax (x) + {\left (b m^{2} p \log \relax (c) + a m^{2} p\right )} \log \relax (x)^{2} - 2 \, {\left (b m n p \log \relax (x) + b m n \log \relax (f)\right )} {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) + {\left (b m^{2} n p \log \relax (x)^{2} + 2 \, b m^{2} n \log \relax (f) \log \relax (x)\right )} \log \left (e x^{m} + d\right ) - {\left (b m^{2} n p \log \relax (x)^{2} + 2 \, b m^{2} n \log \relax (f) \log \relax (x)\right )} \log \left (\frac {e x^{m} + d}{d}\right )}{2 \, m^{2}} \]
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 \[ \int \frac {{\left (b \log \left ({\left (e x^{m} + d\right )}^{n} c\right ) + a\right )} \log \left (f x^{p}\right )}{x}\,{d x} \]
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 \[ \int \frac {\left (b \ln \left (c \left (e \,x^{m}+d \right )^{n}\right )+a \right ) \ln \left (f \,x^{p}\right )}{x}\, dx \]
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 \[ -\frac {1}{2} \, {\left (b p \log \relax (x)^{2} - 2 \, b \log \relax (f) \log \relax (x) - 2 \, b \log \relax (x) \log \left (x^{p}\right )\right )} \log \left ({\left (e x^{m} + d\right )}^{n}\right ) - \int -\frac {2 \, b d \log \relax (c) \log \relax (f) + 2 \, a d \log \relax (f) + {\left (b e m n p \log \relax (x)^{2} - 2 \, b e m n \log \relax (f) \log \relax (x) + 2 \, b e \log \relax (c) \log \relax (f) + 2 \, a e \log \relax (f)\right )} x^{m} + 2 \, {\left (b d \log \relax (c) + a d - {\left (b e m n \log \relax (x) - b e \log \relax (c) - a e\right )} x^{m}\right )} \log \left (x^{p}\right )}{2 \, {\left (e x x^{m} + d x\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (f\,x^p\right )\,\left (a+b\,\ln \left (c\,{\left (d+e\,x^m\right )}^n\right )\right )}{x} \,d x \]
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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