Optimal. Leaf size=114 \[ \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{2 b n}-\frac {r \text {Li}_2\left (-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m}-\frac {r \log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n}+\frac {b n r \text {Li}_3\left (-\frac {f x^m}{e}\right )}{m^2} \]
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Rubi [A] time = 0.19, antiderivative size = 114, 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 = {2375, 2337, 2374, 6589} \[ -\frac {r \text {PolyLog}\left (2,-\frac {f x^m}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{m}+\frac {b n r \text {PolyLog}\left (3,-\frac {f x^m}{e}\right )}{m^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{2 b n}-\frac {r \log \left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{2 b n} \]
Antiderivative was successfully verified.
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Rule 2337
Rule 2374
Rule 2375
Rule 6589
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (d \left (e+f x^m\right )^r\right )}{x} \, dx &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{2 b n}-\frac {(f m r) \int \frac {x^{-1+m} \left (a+b \log \left (c x^n\right )\right )^2}{e+f x^m} \, dx}{2 b n}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{2 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^m}{e}\right )}{2 b n}+r \int \frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {f x^m}{e}\right )}{x} \, dx\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{2 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^m}{e}\right )}{2 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^m}{e}\right )}{m}+\frac {(b n r) \int \frac {\text {Li}_2\left (-\frac {f x^m}{e}\right )}{x} \, dx}{m}\\ &=\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (d \left (e+f x^m\right )^r\right )}{2 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {f x^m}{e}\right )}{2 b n}-\frac {r \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f x^m}{e}\right )}{m}+\frac {b n r \text {Li}_3\left (-\frac {f x^m}{e}\right )}{m^2}\\ \end {align*}
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Mathematica [B] time = 0.17, size = 277, normalized size = 2.43 \[ \frac {r \text {Li}_2\left (\frac {f x^m}{e}+1\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{m}+\frac {a \log \left (-\frac {f x^m}{e}\right ) \log \left (d \left (e+f x^m\right )^r\right )}{m}+b \log (x) \log \left (c x^n\right ) \log \left (d \left (e+f x^m\right )^r\right )-b r \log (x) \log \left (c x^n\right ) \log \left (e+f x^m\right )+\frac {b r \log \left (c x^n\right ) \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac {1}{2} b n \log ^2(x) \log \left (d \left (e+f x^m\right )^r\right )+\frac {b n r \text {Li}_3\left (-\frac {e x^{-m}}{f}\right )}{m^2}+\frac {b n r \log (x) \text {Li}_2\left (-\frac {e x^{-m}}{f}\right )}{m}-\frac {1}{2} b n r \log ^2(x) \log \left (\frac {e x^{-m}}{f}+1\right )+b n r \log ^2(x) \log \left (e+f x^m\right )-\frac {b n r \log (x) \log \left (-\frac {f x^m}{e}\right ) \log \left (e+f x^m\right )}{m}-\frac {1}{6} b m n r \log ^3(x) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.70, size = 173, normalized size = 1.52 \[ \frac {b m^{2} n \log \relax (d) \log \relax (x)^{2} + 2 \, b n r {\rm polylog}\left (3, -\frac {f x^{m}}{e}\right ) + 2 \, {\left (b m^{2} \log \relax (c) + a m^{2}\right )} \log \relax (d) \log \relax (x) - 2 \, {\left (b m n r \log \relax (x) + b m r \log \relax (c) + a m r\right )} {\rm Li}_2\left (-\frac {f x^{m} + e}{e} + 1\right ) + {\left (b m^{2} n r \log \relax (x)^{2} + 2 \, {\left (b m^{2} r \log \relax (c) + a m^{2} r\right )} \log \relax (x)\right )} \log \left (f x^{m} + e\right ) - {\left (b m^{2} n r \log \relax (x)^{2} + 2 \, {\left (b m^{2} r \log \relax (c) + a m^{2} r\right )} \log \relax (x)\right )} \log \left (\frac {f x^{m} + e}{e}\right )}{2 \, m^{2}} \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} \log \left ({\left (f x^{m} + e\right )}^{r} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right ) \ln \left (d \left (f \,x^{m}+e \right )^{r}\right )}{x}\, dx \]
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 \[ -\frac {1}{2} \, {\left (b n \log \relax (x)^{2} - 2 \, b \log \relax (x) \log \left (x^{n}\right ) - 2 \, {\left (b \log \relax (c) + a\right )} \log \relax (x)\right )} \log \left ({\left (f x^{m} + e\right )}^{r}\right ) - \int -\frac {2 \, b e \log \relax (c) \log \relax (d) + 2 \, a e \log \relax (d) + {\left (b f m n r \log \relax (x)^{2} + 2 \, b f \log \relax (c) \log \relax (d) + 2 \, a f \log \relax (d) - 2 \, {\left (b f m r \log \relax (c) + a f m r\right )} \log \relax (x)\right )} x^{m} + 2 \, {\left (b e \log \relax (d) - {\left (b f m r \log \relax (x) - b f \log \relax (d)\right )} x^{m}\right )} \log \left (x^{n}\right )}{2 \, {\left (f x x^{m} + e x\right )}}\,{d x} \]
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 \[ \int \frac {\ln \left (d\,{\left (e+f\,x^m\right )}^r\right )\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]
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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