Optimal. Leaf size=80 \[ -\frac {\log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}+\frac {b n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^2}+\frac {b n \log (d+e x)}{d^2} \]
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Rubi [A] time = 0.16, antiderivative size = 102, normalized size of antiderivative = 1.28, number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2347, 2344, 2301, 2317, 2391, 2314, 31} \[ -\frac {b n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}-\frac {\log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac {b n \log (d+e x)}{d^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2301
Rule 2314
Rule 2317
Rule 2344
Rule 2347
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)^2} \, dx &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d}\\ &=-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}+\frac {\int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^2}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2}+\frac {(b e n) \int \frac {1}{d+e x} \, dx}{d^2}\\ &=-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac {b n \log (d+e x)}{d^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2}+\frac {(b n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2}\\ &=-\frac {e x \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac {b n \log (d+e x)}{d^2}-\frac {\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2}-\frac {b n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 96, normalized size = 1.20 \[ \frac {-2 \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{d+e x}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{b n}-2 b n \text {Li}_2\left (-\frac {e x}{d}\right )-2 b n (\log (x)-\log (d+e x))}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e^{2} x^{3} + 2 \, d e x^{2} + d^{2} x}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{2} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 521, normalized size = 6.51 \[ \frac {b n \ln \left (-\frac {e x}{d}\right ) \ln \left (e x +d \right )}{d^{2}}-\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 \left (e x +d \right ) d}-\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )}{2 d^{2}}+\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e x +d \right )}{2 d^{2}}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (e x +d \right )}{2 d^{2}}-\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (x )}{2 d^{2}}-\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 \left (e x +d \right ) d}-\frac {b n \ln \relax (x )^{2}}{2 d^{2}}-\frac {b n \ln \relax (x )}{d^{2}}+\frac {b \ln \left (x^{n}\right )}{\left (e x +d \right ) d}+\frac {b \ln \relax (x ) \ln \left (x^{n}\right )}{d^{2}}-\frac {b \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{2}}+\frac {b \ln \relax (c ) \ln \relax (x )}{d^{2}}-\frac {b \ln \relax (c ) \ln \left (e x +d \right )}{d^{2}}+\frac {b n \dilog \left (-\frac {e x}{d}\right )}{d^{2}}+\frac {b \ln \relax (c )}{\left (e x +d \right ) d}+\frac {a \ln \relax (x )}{d^{2}}-\frac {a \ln \left (e x +d \right )}{d^{2}}+\frac {a}{\left (e x +d \right ) d}-\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 d^{2}}+\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )}{2 d^{2}}-\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 d^{2}}+\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 \left (e x +d \right ) d}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (x )}{2 d^{2}}+\frac {i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 \left (e x +d \right ) d}+\frac {b n \ln \left (e x +d \right )}{d^{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 \[ a {\left (\frac {1}{d e x + d^{2}} - \frac {\log \left (e x + d\right )}{d^{2}} + \frac {\log \relax (x)}{d^{2}}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e^{2} x^{3} + 2 \, d e x^{2} + d^{2} x}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,{\left (d+e\,x\right )}^2} \,d x \]
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 \[ \int \frac {a + b \log {\left (c x^{n} \right )}}{x \left (d + e x\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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