Optimal. Leaf size=50 \[ \frac{\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 B g i n (b c-a d)} \]
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Rubi [C] time = 0.556831, antiderivative size = 316, normalized size of antiderivative = 6.32, number of steps used = 18, number of rules used = 8, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.186, Rules used = {2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{B n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g i (b c-a d)}+\frac{B n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g i (b c-a d)}+\frac{\log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g i (b c-a d)}-\frac{\log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g i (b c-a d)}-\frac{B n \log ^2(a+b x)}{2 g i (b c-a d)}-\frac{B n \log ^2(c+d x)}{2 g i (b c-a d)}+\frac{B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g i (b c-a d)}+\frac{B n \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g i (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(139 c+139 d x) (a g+b g x)} \, dx &=\int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g (c+d x)}\right ) \, dx\\ &=\frac{b \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{139 (b c-a d) g}-\frac{d \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{139 (b c-a d) g}\\ &=\frac{\log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\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 (a+b x)}{a+b x} \, dx}{139 (b c-a d) g}+\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 (c+d x)}{a+b x} \, dx}{139 (b c-a d) g}\\ &=\frac{\log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac{(B n) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{139 (b c-a d) g}+\frac{(B n) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{139 (b c-a d) g}\\ &=\frac{\log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac{(b B n) \int \frac{\log (a+b x)}{a+b x} \, dx}{139 (b c-a d) g}+\frac{(b B n) \int \frac{\log (c+d x)}{a+b x} \, dx}{139 (b c-a d) g}+\frac{(B d n) \int \frac{\log (a+b x)}{c+d x} \, dx}{139 (b c-a d) g}-\frac{(B d n) \int \frac{\log (c+d x)}{c+d x} \, dx}{139 (b c-a d) g}\\ &=\frac{\log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}+\frac{B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{139 (b c-a d) g}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}+\frac{B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}-\frac{(B n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{139 (b c-a d) g}-\frac{(B n) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{139 (b c-a d) g}-\frac{(b B n) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{139 (b c-a d) g}-\frac{(B d n) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{139 (b c-a d) g}\\ &=-\frac{B n \log ^2(a+b x)}{278 (b c-a d) g}+\frac{\log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}+\frac{B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{139 (b c-a d) g}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac{B n \log ^2(c+d x)}{278 (b c-a d) g}+\frac{B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}-\frac{(B n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{139 (b c-a d) g}-\frac{(B n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{139 (b c-a d) g}\\ &=-\frac{B n \log ^2(a+b x)}{278 (b c-a d) g}+\frac{\log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{139 (b c-a d) g}+\frac{B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{139 (b c-a d) g}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{139 (b c-a d) g}-\frac{B n \log ^2(c+d x)}{278 (b c-a d) g}+\frac{B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}+\frac{B n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{139 (b c-a d) g}+\frac{B n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{139 (b c-a d) g}\\ \end{align*}
Mathematica [C] time = 0.103702, size = 219, normalized size = 4.38 \[ \frac{2 B n \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )+2 B n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+2 A \log (a+b x)+2 B \log (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 B \log (c+d x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+2 B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )+2 B n \log (c+d x) \log \left (\frac{d (a+b x)}{a d-b c}\right )-B n \log ^2(a+b x)-2 A \log (c+d x)-B n \log ^2(c+d x)}{2 g i (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.758, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) \left ( dix+ci \right ) } \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 [B] time = 1.16397, size = 236, normalized size = 4.72 \begin{align*} B{\left (\frac{\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac{\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) - \frac{{\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} B n}{2 \,{\left (b c g i - a d g i\right )}} + A{\left (\frac{\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac{\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.513175, size = 169, normalized size = 3.38 \begin{align*} \frac{B n \log \left (\frac{b x + a}{d x + c}\right )^{2} + 2 \, B \log \left (e\right ) \log \left (\frac{b x + a}{d x + c}\right ) + 2 \, A \log \left (\frac{b x + a}{d x + c}\right )}{2 \,{\left (b c - a d\right )} g i} \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 [B] time = 1.30973, size = 143, normalized size = 2.86 \begin{align*} -\frac{B i n \log \left (\frac{b x + a}{d x + c}\right )^{2}}{2 \,{\left (b c g - a d g\right )}} - \frac{{\left (A i + B i\right )} \log \left ({\left | \frac{2 \, b d x + b c + a d -{\left | -b c + a d \right |}}{2 \, b d x + b c + a d +{\left | -b c + a d \right |}} \right |}\right )}{g{\left | -b c + a d \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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