Optimal. Leaf size=69 \[ -\frac {d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {a x}{e}+\frac {b x \log \left (c x^n\right )}{e}-\frac {b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2}-\frac {b n x}{e} \]
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Rubi [A] time = 0.09, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {43, 2351, 2295, 2317, 2391} \[ -\frac {b d n \text {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2}-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}+\frac {a x}{e}+\frac {b x \log \left (c x^n\right )}{e}-\frac {b n x}{e} \]
Antiderivative was successfully verified.
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Rule 43
Rule 2295
Rule 2317
Rule 2351
Rule 2391
Rubi steps
\begin {align*} \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{d+e x} \, dx &=\int \left (\frac {a+b \log \left (c x^n\right )}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )}{e (d+e x)}\right ) \, dx\\ &=\frac {\int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e}-\frac {d \int \frac {a+b \log \left (c x^n\right )}{d+e x} \, dx}{e}\\ &=\frac {a x}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^2}+\frac {b \int \log \left (c x^n\right ) \, dx}{e}+\frac {(b d n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^2}\\ &=\frac {a x}{e}-\frac {b n x}{e}+\frac {b x \log \left (c x^n\right )}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 66, normalized size = 0.96 \[ \frac {-a d \log \left (\frac {e x}{d}+1\right )+a e x+b \log \left (c x^n\right ) \left (e x-d \log \left (\frac {e x}{d}+1\right )\right )-b d n \text {Li}_2\left (-\frac {e x}{d}\right )-b e n x}{e^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b x \log \left (c x^{n}\right ) + a x}{e x + d}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x}{e x + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.23, size = 343, normalized size = 4.97 \[ \frac {i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (e x +d \right )}{2 e^{2}}-\frac {i \pi b d \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 e^{2}}-\frac {i \pi b d \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (e x +d \right )}{2 e^{2}}+\frac {i \pi b d \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (e x +d \right )}{2 e^{2}}-\frac {i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )}{2 e}+\frac {i \pi b x \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 e}+\frac {i \pi b x \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2}}{2 e}-\frac {i \pi b x \mathrm {csgn}\left (i c \,x^{n}\right )^{3}}{2 e}+\frac {b d n \ln \left (-\frac {e x}{d}\right ) \ln \left (e x +d \right )}{e^{2}}+\frac {b d n \dilog \left (-\frac {e x}{d}\right )}{e^{2}}-\frac {b d \ln \relax (c ) \ln \left (e x +d \right )}{e^{2}}-\frac {b d \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{e^{2}}-\frac {b n x}{e}+\frac {b x \ln \relax (c )}{e}+\frac {b x \ln \left (x^{n}\right )}{e}-\frac {a d \ln \left (e x +d \right )}{e^{2}}+\frac {a x}{e}-\frac {b d n}{e^{2}} \]
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 \[ a {\left (\frac {x}{e} - \frac {d \log \left (e x + d\right )}{e^{2}}\right )} + b \int \frac {x \log \relax (c) + x \log \left (x^{n}\right )}{e x + d}\,{d x} \]
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 \[ \int \frac {x\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 13.68, size = 144, normalized size = 2.09 \[ - \frac {a d \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right )}{e} + \frac {a x}{e} + \frac {b d n \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\begin {cases} \log {\relax (d )} \log {\relax (x )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \left |{x}\right | < 1 \\- \log {\relax (d )} \log {\left (\frac {1}{x} \right )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {for}\: \frac {1}{\left |{x}\right |} < 1 \\- {G_{2, 2}^{2, 0}\left (\begin {matrix} & 1, 1 \\0, 0 & \end {matrix} \middle | {x} \right )} \log {\relax (d )} + {G_{2, 2}^{0, 2}\left (\begin {matrix} 1, 1 & \\ & 0, 0 \end {matrix} \middle | {x} \right )} \log {\relax (d )} - \operatorname {Li}_{2}\left (\frac {e x e^{i \pi }}{d}\right ) & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right )}{e} - \frac {b d \left (\begin {cases} \frac {x}{d} & \text {for}\: e = 0 \\\frac {\log {\left (d + e x \right )}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )}}{e} - \frac {b n x}{e} + \frac {b x \log {\left (c x^{n} \right )}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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