Optimal. Leaf size=141 \[ \frac{B i n (b c-a d) \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right )}{b^2 g}-\frac{i (b c-a d) \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A-B n\right )}{b^2 g}+\frac{i (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b g} \]
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Rubi [A] time = 0.344796, antiderivative size = 223, normalized size of antiderivative = 1.58, number of steps used = 13, number of rules used = 10, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.244, Rules used = {2528, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B i n (b c-a d) \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}+\frac{i (b c-a d) \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g}+\frac{B d i (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}-\frac{B i n (b c-a d) \log ^2(a+b x)}{2 b^2 g}+\frac{B i n (b c-a d) \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{B i n (b c-a d) \log (c+d x)}{b^2 g}+\frac{A d i x}{b g} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2486
Rule 31
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{(112 c+112 d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a g+b g x} \, dx &=\int \left (\frac{112 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b g}+\frac{112 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b g (a+b x)}\right ) \, dx\\ &=\frac{(112 d) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b g}+\frac{(112 (b c-a d)) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b g}\\ &=\frac{112 A d x}{b g}+\frac{112 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}+\frac{(112 B d) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b g}-\frac{(112 B (b c-a d) n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 g}\\ &=\frac{112 A d x}{b g}+\frac{112 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{112 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}-\frac{(112 B (b c-a d) n) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^2 g}-\frac{(112 B d (b c-a d) n) \int \frac{1}{c+d x} \, dx}{b^2 g}\\ &=\frac{112 A d x}{b g}+\frac{112 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{112 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}-\frac{112 B (b c-a d) n \log (c+d x)}{b^2 g}-\frac{(112 B (b c-a d) n) \int \frac{\log (a+b x)}{a+b x} \, dx}{b g}+\frac{(112 B d (b c-a d) n) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^2 g}\\ &=\frac{112 A d x}{b g}+\frac{112 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{112 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}-\frac{112 B (b c-a d) n \log (c+d x)}{b^2 g}+\frac{112 B (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{(112 B (b c-a d) n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g}-\frac{(112 B (b c-a d) n) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g}\\ &=\frac{112 A d x}{b g}-\frac{56 B (b c-a d) n \log ^2(a+b x)}{b^2 g}+\frac{112 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{112 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}-\frac{112 B (b c-a d) n \log (c+d x)}{b^2 g}+\frac{112 B (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}-\frac{(112 B (b c-a d) n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g}\\ &=\frac{112 A d x}{b g}-\frac{56 B (b c-a d) n \log ^2(a+b x)}{b^2 g}+\frac{112 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^2 g}+\frac{112 (b c-a d) \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 g}-\frac{112 B (b c-a d) n \log (c+d x)}{b^2 g}+\frac{112 B (b c-a d) n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^2 g}+\frac{112 B (b c-a d) n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^2 g}\\ \end{align*}
Mathematica [A] time = 0.123463, size = 172, normalized size = 1.22 \[ \frac{i \left (2 B n (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+2 (b c-a d) \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+B n \log \left (\frac{b (c+d x)}{b c-a d}\right )+A\right )+2 \left (B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+B n (a d-b c) \log (c+d x)+A b d x\right )+B n (a d-b c) \log ^2(a+b x)\right )}{2 b^2 g} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.619, size = 0, normalized size = 0. \begin{align*} \int{\frac{dix+ci}{bgx+ag} \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 [A] time = 2.89133, size = 373, normalized size = 2.65 \begin{align*} A d i{\left (\frac{x}{b g} - \frac{a \log \left (b x + a\right )}{b^{2} g}\right )} - \frac{B c i n \log \left (d x + c\right )}{b g} + \frac{A c i \log \left (b g x + a g\right )}{b g} + \frac{{\left (b c i n - a d i n\right )}{\left (\log \left (b x + a\right ) \log \left (\frac{b d x + a d}{b c - a d} + 1\right ) +{\rm Li}_2\left (-\frac{b d x + a d}{b c - a d}\right )\right )} B}{b^{2} g} + \frac{2 \, B b d i x \log \left (e\right ) -{\left (b c i n - a d i n\right )} B \log \left (b x + a\right )^{2} + 2 \,{\left (b c i \log \left (e\right ) +{\left (i n - i \log \left (e\right )\right )} a d\right )} B \log \left (b x + a\right ) + 2 \,{\left (B b d i x +{\left (b c i - a d i\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \,{\left (B b d i x +{\left (b c i - a d i\right )} B \log \left (b x + a\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, b^{2} g} \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{A d i x + A c i +{\left (B d i x + B c i\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{b g x + a g}, 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{{\left (d i x + c i\right )}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}}{b g x + a g}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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