Optimal. Leaf size=39 \[ \frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e} \]
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Rubi [A] time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2314, 31} \[ \frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2314
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx &=\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {(b n) \int \frac {1}{d+e x} \, dx}{d}\\ &=\frac {x \left (a+b \log \left (c x^n\right )\right )}{d (d+e x)}-\frac {b n \log (d+e x)}{d e}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 41, normalized size = 1.05 \[ \frac {\frac {b n (\log (x)-\log (d+e x))}{d}-\frac {a+b \log \left (c x^n\right )}{d+e x}}{e} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 51, normalized size = 1.31 \[ \frac {b e n x \log \relax (x) - b d \log \relax (c) - a d - {\left (b e n x + b d n\right )} \log \left (e x + d\right )}{d e^{2} x + d^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 58, normalized size = 1.49 \[ -\frac {b n x e \log \left (x e + d\right ) - b n x e \log \relax (x) + b d n \log \left (x e + d\right ) + b d \log \relax (c) + a d}{d x e^{2} + d^{2} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.20, size = 173, normalized size = 4.44 \[ -\frac {b \ln \left (x^{n}\right )}{\left (e x +d \right ) e}-\frac {-i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )+i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}+i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3}-2 b e n x \ln \left (-x \right )+2 b e n x \ln \left (e x +d \right )-2 b d n \ln \left (-x \right )+2 b d n \ln \left (e x +d \right )+2 b d \ln \relax (c )+2 a d}{2 \left (e x +d \right ) d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 63, normalized size = 1.62 \[ -b n {\left (\frac {\log \left (e x + d\right )}{d e} - \frac {\log \relax (x)}{d e}\right )} - \frac {b \log \left (c x^{n}\right )}{e^{2} x + d e} - \frac {a}{e^{2} x + d e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.56, size = 54, normalized size = 1.38 \[ -\frac {a}{x\,e^2+d\,e}-\frac {b\,\ln \left (c\,x^n\right )}{e\,\left (d+e\,x\right )}-\frac {2\,b\,n\,\mathrm {atanh}\left (\frac {2\,e\,x}{d}+1\right )}{d\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.35, size = 187, normalized size = 4.79 \[ \begin {cases} \tilde {\infty } \left (- \frac {a}{x} - \frac {b n \log {\relax (x )}}{x} - \frac {b n}{x} - \frac {b \log {\relax (c )}}{x}\right ) & \text {for}\: d = 0 \wedge e = 0 \\\frac {a x + b n x \log {\relax (x )} - b n x + b x \log {\relax (c )}}{d^{2}} & \text {for}\: e = 0 \\\frac {- \frac {a}{x} - \frac {b n \log {\relax (x )}}{x} - \frac {b n}{x} - \frac {b \log {\relax (c )}}{x}}{e^{2}} & \text {for}\: d = 0 \\- \frac {a d}{d^{2} e + d e^{2} x} - \frac {b d n \log {\left (\frac {d}{e} + x \right )}}{d^{2} e + d e^{2} x} + \frac {b e n x \log {\relax (x )}}{d^{2} e + d e^{2} x} - \frac {b e n x \log {\left (\frac {d}{e} + x \right )}}{d^{2} e + d e^{2} x} + \frac {b e x \log {\relax (c )}}{d^{2} e + d e^{2} x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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