Optimal. Leaf size=107 \[ -\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}+\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}-\frac {b n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{f}+\frac {b n \text {Li}_2\left (\frac {e x}{d}+1\right )}{f} \]
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Rubi [A] time = 0.14, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {36, 29, 31, 2416, 2394, 2315, 2393, 2391} \[ -\frac {b n \text {PolyLog}\left (2,-\frac {g (d+e x)}{e f-d g}\right )}{f}+\frac {b n \text {PolyLog}\left (2,\frac {e x}{d}+1\right )}{f}-\frac {\log \left (\frac {e (f+g x)}{e f-d g}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}+\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2315
Rule 2391
Rule 2393
Rule 2394
Rule 2416
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{x (f+g x)} \, dx &=\int \left (\frac {a+b \log \left (c (d+e x)^n\right )}{f x}-\frac {g \left (a+b \log \left (c (d+e x)^n\right )\right )}{f (f+g x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c (d+e x)^n\right )}{x} \, dx}{f}-\frac {g \int \frac {a+b \log \left (c (d+e x)^n\right )}{f+g x} \, dx}{f}\\ &=\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{f}-\frac {(b e n) \int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx}{f}+\frac {(b e n) \int \frac {\log \left (\frac {e (f+g x)}{e f-d g}\right )}{d+e x} \, dx}{f}\\ &=\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{f}+\frac {b n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f}+\frac {(b n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {g x}{e f-d g}\right )}{x} \, dx,x,d+e x\right )}{f}\\ &=\frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{f}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e (f+g x)}{e f-d g}\right )}{f}-\frac {b n \text {Li}_2\left (-\frac {g (d+e x)}{e f-d g}\right )}{f}+\frac {b n \text {Li}_2\left (1+\frac {e x}{d}\right )}{f}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 85, normalized size = 0.79 \[ \frac {\left (\log \left (-\frac {e x}{d}\right )-\log \left (\frac {e (f+g x)}{e f-d g}\right )\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-b n \text {Li}_2\left (\frac {g (d+e x)}{d g-e f}\right )+b n \text {Li}_2\left (\frac {e x}{d}+1\right )}{f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{g x^{2} + f x}, x\right ) \]
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 \[ \int \frac {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}{{\left (g x + f\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.25, size = 455, normalized size = 4.25 \[ -\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \ln \relax (x )}{2 f}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{2 f}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \relax (x )}{2 f}-\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right )}{2 f}+\frac {i \pi b \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \relax (x )}{2 f}-\frac {i \pi b \,\mathrm {csgn}\left (i \left (e x +d \right )^{n}\right ) \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2} \ln \left (g x +f \right )}{2 f}-\frac {i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \relax (x )}{2 f}+\frac {i \pi b \mathrm {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3} \ln \left (g x +f \right )}{2 f}-\frac {b n \ln \relax (x ) \ln \left (\frac {e x +d}{d}\right )}{f}+\frac {b n \ln \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right ) \ln \left (g x +f \right )}{f}+\frac {b n \dilog \left (\frac {d g -e f +\left (g x +f \right ) e}{d g -e f}\right )}{f}-\frac {b n \dilog \left (\frac {e x +d}{d}\right )}{f}+\frac {b \ln \relax (c ) \ln \relax (x )}{f}-\frac {b \ln \relax (c ) \ln \left (g x +f \right )}{f}+\frac {b \ln \relax (x ) \ln \left (\left (e x +d \right )^{n}\right )}{f}-\frac {b \ln \left (\left (e x +d \right )^{n}\right ) \ln \left (g x +f \right )}{f}+\frac {a \ln \relax (x )}{f}-\frac {a \ln \left (g x +f \right )}{f} \]
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 \[ -a {\left (\frac {\log \left (g x + f\right )}{f} - \frac {\log \relax (x)}{f}\right )} + b \int \frac {\log \left ({\left (e x + d\right )}^{n}\right ) + \log \relax (c)}{g x^{2} + f x}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{x\,\left (f+g\,x\right )} \,d x \]
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 \[ \int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{x \left (f + g x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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