Optimal. Leaf size=80 \[ -\frac{B n \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{d i}-\frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d i} \]
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Rubi [A] time = 0.20157, antiderivative size = 128, normalized size of antiderivative = 1.6, number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {2524, 2418, 2394, 2393, 2391, 2390, 12, 2301} \[ -\frac{B n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d i}+\frac{\log (c i+d i x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d i}-\frac{B n \log (c i+d i x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d i}+\frac{B n \log ^2(i (c+d x))}{2 d i} \]
Antiderivative was successfully verified.
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Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rule 2390
Rule 12
Rule 2301
Rubi steps
\begin{align*} \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{138 c+138 d x} \, dx &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}-\frac{(B n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (138 c+138 d x)}{a+b x} \, dx}{138 d}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}-\frac{(B n) \int \left (\frac{b \log (138 c+138 d x)}{a+b x}-\frac{d \log (138 c+138 d x)}{c+d x}\right ) \, dx}{138 d}\\ &=\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}+\frac{1}{138} (B n) \int \frac{\log (138 c+138 d x)}{c+d x} \, dx-\frac{(b B n) \int \frac{\log (138 c+138 d x)}{a+b x} \, dx}{138 d}\\ &=-\frac{B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (138 c+138 d x)}{138 d}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}+(B n) \int \frac{\log \left (\frac{138 d (a+b x)}{-138 b c+138 a d}\right )}{138 c+138 d x} \, dx+\frac{(B n) \operatorname{Subst}\left (\int \frac{138 \log (x)}{x} \, dx,x,138 c+138 d x\right )}{19044 d}\\ &=-\frac{B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (138 c+138 d x)}{138 d}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}+\frac{(B n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,138 c+138 d x\right )}{138 d}+\frac{(B n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-138 b c+138 a d}\right )}{x} \, dx,x,138 c+138 d x\right )}{138 d}\\ &=\frac{B n \log ^2(138 (c+d x))}{276 d}-\frac{B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (138 c+138 d x)}{138 d}+\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}-\frac{B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{138 d}\\ \end{align*}
Mathematica [A] time = 0.0322407, size = 101, normalized size = 1.26 \[ \frac{\log (i (c+d x)) \left (2 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 B n \log \left (\frac{d (a+b x)}{a d-b c}\right )+2 A+B n \log (i (c+d x))\right )-2 B n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{2 d i} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.611, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dix+ci} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, B{\left (\frac{2 \, n \log \left (b x + a\right ) \log \left (d x + c\right ) - n \log \left (d x + c\right )^{2} - 2 \, \log \left (d x + c\right ) \log \left ({\left (b x + a\right )}^{n}\right ) + 2 \, \log \left (d x + c\right ) \log \left ({\left (d x + c\right )}^{n}\right )}{d i} - 2 \, \int \frac{n \log \left (b x + a\right ) + \log \left (e\right )}{d i x + c i}\,{d x}\right )} + \frac{A \log \left (d i x + c i\right )}{d i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}{d i x + c i}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}{d i x + c i}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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