Optimal. Leaf size=40 \[ \frac {b n \text {Li}_3\left (-d f x^m\right )}{m^2}-\frac {\text {Li}_2\left (-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m} \]
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Rubi [A] time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2374, 6589} \[ \frac {b n \text {PolyLog}\left (3,-d f x^m\right )}{m^2}-\frac {\text {PolyLog}\left (2,-d f x^m\right ) \left (a+b \log \left (c x^n\right )\right )}{m} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (\frac {1}{d}+f x^m\right )\right )}{x} \, dx &=-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^m\right )}{m}+\frac {(b n) \int \frac {\text {Li}_2\left (-d f x^m\right )}{x} \, dx}{m}\\ &=-\frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-d f x^m\right )}{m}+\frac {b n \text {Li}_3\left (-d f x^m\right )}{m^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 52, normalized size = 1.30 \[ -\frac {a \text {Li}_2\left (-d f x^m\right )}{m}-\frac {b \log \left (c x^n\right ) \text {Li}_2\left (-d f x^m\right )}{m}+\frac {b n \text {Li}_3\left (-d f x^m\right )}{m^2} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.86, size = 42, normalized size = 1.05 \[ \frac {b n {\rm polylog}\left (3, -d f x^{m}\right ) - {\left (b m n \log \relax (x) + b m \log \relax (c) + a m\right )} {\rm Li}_2\left (-d f x^{m}\right )}{m^{2}} \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{m} + \frac {1}{d}\right )} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.94, size = 308, normalized size = 7.70 \[ -\frac {b n \ln \relax (x )^{2} \ln \left (\left (f \,x^{m}+\frac {1}{d}\right ) d \right )}{2}+\frac {b n \ln \relax (x )^{2} \ln \left (d f \,x^{m}+1\right )}{2}+\frac {i \pi b \dilog \left (d f \,x^{m}+1\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 m}-\frac {i \pi b \dilog \left (d f \,x^{m}+1\right ) \mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 m}-\frac {i \pi b \dilog \left (d f \,x^{m}+1\right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 m}+\frac {i \pi b \dilog \left (d f \,x^{m}+1\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 m}+b \ln \relax (x ) \ln \left (x^{n}\right ) \ln \left (\left (f \,x^{m}+\frac {1}{d}\right ) d \right )-b \ln \relax (x ) \ln \left (x^{n}\right ) \ln \left (d f \,x^{m}+1\right )+\frac {b n \dilog \left (d f \,x^{m}+1\right ) \ln \relax (x )}{m}-\frac {b n \polylog \left (2, -d f \,x^{m}\right ) \ln \relax (x )}{m}-\frac {b \dilog \left (d f \,x^{m}+1\right ) \ln \relax (c )}{m}-\frac {b \dilog \left (d f \,x^{m}+1\right ) \ln \left (x^{n}\right )}{m}-\frac {a \dilog \left (d f \,x^{m}+1\right )}{m}+\frac {b n \polylog \left (3, -d f \,x^{m}\right )}{m^{2}} \]
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 \[ -\frac {1}{2} \, {\left (b n \log \relax (x)^{2} - 2 \, b \log \relax (x) \log \left (x^{n}\right ) - 2 \, {\left (b \log \relax (c) + a\right )} \log \relax (x)\right )} \log \left (d f x^{m} + 1\right ) - \int \frac {2 \, b d f m x^{m} \log \relax (x) \log \left (x^{n}\right ) - {\left (b d f m n \log \relax (x)^{2} - 2 \, {\left (b d f m \log \relax (c) + a d f m\right )} \log \relax (x)\right )} x^{m}}{2 \, {\left (d f x x^{m} + x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\ln \left (d\,\left (f\,x^m+\frac {1}{d}\right )\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]
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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