Optimal. Leaf size=74 \[ \frac {e \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac {a+b \log \left (c x^n\right )}{d x}-\frac {b e n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^2}-\frac {b n}{d x} \]
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Rubi [A] time = 0.14, antiderivative size = 95, normalized size of antiderivative = 1.28, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {44, 2351, 2304, 2301, 2317, 2391} \[ \frac {b e n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{d^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac {e \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2}-\frac {a+b \log \left (c x^n\right )}{d x}-\frac {b n}{d x} \]
Antiderivative was successfully verified.
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Rule 44
Rule 2301
Rule 2304
Rule 2317
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{d x^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^2 x}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d}-\frac {e \int \frac {a+b \log \left (c x^n\right )}{x} \, dx}{d^2}+\frac {e^2 \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^2}\\ &=-\frac {b n}{d x}-\frac {a+b \log \left (c x^n\right )}{d x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac {e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2}-\frac {(b e n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{d^2}\\ &=-\frac {b n}{d x}-\frac {a+b \log \left (c x^n\right )}{d x}-\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^2 n}+\frac {e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{d^2}+\frac {b e n \text {Li}_2\left (-\frac {e x}{d}\right )}{d^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 88, normalized size = 1.19 \[ -\frac {-2 e \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 d \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{b n}-2 b e n \text {Li}_2\left (-\frac {e x}{d}\right )+\frac {2 b d n}{x}}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (c x^{n}\right ) + a}{e x^{3} + d x^{2}}, x\right ) \]
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 \[ \int \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.19, size = 504, normalized size = 6.81 \[ -\frac {b e n \dilog \left (-\frac {e x}{d}\right )}{d^{2}}+\frac {b e \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{d^{2}}-\frac {b e \ln \relax (x ) \ln \left (x^{n}\right )}{d^{2}}+\frac {b e n \ln \relax (x )^{2}}{2 d^{2}}-\frac {b e \ln \relax (c ) \ln \relax (x )}{d^{2}}+\frac {b e \ln \relax (c ) \ln \left (e x +d \right )}{d^{2}}+\frac {i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 d x}-\frac {b e n \ln \left (-\frac {e x}{d}\right ) \ln \left (e x +d \right )}{d^{2}}-\frac {a}{d x}-\frac {b \ln \left (x^{n}\right )}{d x}-\frac {i \pi b e \,\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 e \,\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 e \,\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 e \,\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 {a e \ln \relax (x )}{d^{2}}+\frac {a e \ln \left (e x +d \right )}{d^{2}}-\frac {b \ln \relax (c )}{d x}+\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 d x}+\frac {i \pi b e \,\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 {b n}{d x}+\frac {i \pi b e \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (x )}{2 d^{2}}-\frac {i \pi b e \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 c \,x^{n}\right )^{2}}{2 d x}-\frac {i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 d x}-\frac {i \pi b e \,\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}} \]
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 \[ a {\left (\frac {e \log \left (e x + d\right )}{d^{2}} - \frac {e \log \relax (x)}{d^{2}} - \frac {1}{d x}\right )} + b \int \frac {\log \relax (c) + \log \left (x^{n}\right )}{e x^{3} + d x^{2}}\,{d x} \]
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 \[ \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^2\,\left (d+e\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 61.19, size = 197, normalized size = 2.66 \[ - \frac {a}{d x} + \frac {a e^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{d^{2}} - \frac {a e \log {\relax (x )}}{d^{2}} - \frac {b n}{d x} - \frac {b \log {\left (c x^{n} \right )}}{d x} - \frac {b e^{2} n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} \log {\relax (d )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (d )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (d )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (d )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{d^{2}} + \frac {b e^{2} \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{d^{2}} + \frac {b e n \log {\relax (x )}^{2}}{2 d^{2}} - \frac {b e \log {\relax (x )} \log {\left (c x^{n} \right )}}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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