Optimal. Leaf size=322 \[ \frac{2 B g n \text{PolyLog}\left (2,-\frac{b (f x+g)}{a f-b g}\right )}{f^3}-\frac{2 B g n \text{PolyLog}\left (2,-\frac{d (f x+g)}{c f-d g}\right )}{f^3}-\frac{g^2 (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^2 (f x+g) (a f-b g)}-\frac{2 g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^3}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^2}+\frac{B g^2 n (b c-a d) \log \left (\frac{f x+g}{c+d x}\right )}{f^2 (a f-b g) (c f-d g)}-\frac{B n (b c-a d) \log (c+d x)}{b d f^2}+\frac{2 B g n \log (f x+g) \log \left (\frac{f (a+b x)}{a f-b g}\right )}{f^3}+\frac{A x}{f^2}-\frac{2 B g n \log (f x+g) \log \left (\frac{f (c+d x)}{c f-d g}\right )}{f^3} \]
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Rubi [A] time = 0.493759, antiderivative size = 352, normalized size of antiderivative = 1.09, number of steps used = 18, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2528, 2486, 31, 2525, 12, 72, 2524, 2418, 2394, 2393, 2391} \[ \frac{2 B g n \text{PolyLog}\left (2,-\frac{b (f x+g)}{a f-b g}\right )}{f^3}-\frac{2 B g n \text{PolyLog}\left (2,-\frac{d (f x+g)}{c f-d g}\right )}{f^3}-\frac{g^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^3 (f x+g)}-\frac{2 g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f^3}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^2}+\frac{B g^2 n (b c-a d) \log (f x+g)}{f^2 (a f-b g) (c f-d g)}-\frac{B n (b c-a d) \log (c+d x)}{b d f^2}-\frac{b B g^2 n \log (a+b x)}{f^3 (a f-b g)}+\frac{2 B g n \log (f x+g) \log \left (\frac{f (a+b x)}{a f-b g}\right )}{f^3}+\frac{A x}{f^2}+\frac{B d g^2 n \log (c+d x)}{f^3 (c f-d g)}-\frac{2 B g n \log (f x+g) \log \left (\frac{f (c+d x)}{c f-d g}\right )}{f^3} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2486
Rule 31
Rule 2525
Rule 12
Rule 72
Rule 2524
Rule 2418
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 )}{\left (f+\frac{g}{x}\right )^2} \, dx &=\int \left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f^2}+\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^2 (g+f x)^2}-\frac{2 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^2 (g+f x)}\right ) \, dx\\ &=\frac{\int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{f^2}-\frac{(2 g) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{g+f x} \, dx}{f^2}+\frac{g^2 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(g+f x)^2} \, dx}{f^2}\\ &=\frac{A x}{f^2}-\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)}-\frac{2 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^3}+\frac{B \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{f^2}+\frac{(2 B g n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (g+f x)}{a+b x} \, dx}{f^3}+\frac{\left (B g^2 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x) (g+f x)} \, dx}{f^3}\\ &=\frac{A x}{f^2}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^2}-\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)}-\frac{2 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^3}-\frac{(B (b c-a d) n) \int \frac{1}{c+d x} \, dx}{b f^2}+\frac{(2 B g n) \int \left (\frac{b \log (g+f x)}{a+b x}-\frac{d \log (g+f x)}{c+d x}\right ) \, dx}{f^3}+\frac{\left (B (b c-a d) g^2 n\right ) \int \frac{1}{(a+b x) (c+d x) (g+f x)} \, dx}{f^3}\\ &=\frac{A x}{f^2}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^2}-\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^2}-\frac{2 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^3}+\frac{(2 b B g n) \int \frac{\log (g+f x)}{a+b x} \, dx}{f^3}-\frac{(2 B d g n) \int \frac{\log (g+f x)}{c+d x} \, dx}{f^3}+\frac{\left (B (b c-a d) g^2 n\right ) \int \left (\frac{b^2}{(b c-a d) (-a f+b g) (a+b x)}+\frac{d^2}{(b c-a d) (c f-d g) (c+d x)}+\frac{f^2}{(a f-b g) (c f-d g) (g+f x)}\right ) \, dx}{f^3}\\ &=\frac{A x}{f^2}-\frac{b B g^2 n \log (a+b x)}{f^3 (a f-b g)}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^2}-\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^2}+\frac{B d g^2 n \log (c+d x)}{f^3 (c f-d g)}+\frac{B (b c-a d) g^2 n \log (g+f x)}{f^2 (a f-b g) (c f-d g)}+\frac{2 B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^3}-\frac{2 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^3}-\frac{2 B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^3}-\frac{(2 B g n) \int \frac{\log \left (\frac{f (a+b x)}{a f-b g}\right )}{g+f x} \, dx}{f^2}+\frac{(2 B g n) \int \frac{\log \left (\frac{f (c+d x)}{c f-d g}\right )}{g+f x} \, dx}{f^2}\\ &=\frac{A x}{f^2}-\frac{b B g^2 n \log (a+b x)}{f^3 (a f-b g)}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^2}-\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^2}+\frac{B d g^2 n \log (c+d x)}{f^3 (c f-d g)}+\frac{B (b c-a d) g^2 n \log (g+f x)}{f^2 (a f-b g) (c f-d g)}+\frac{2 B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^3}-\frac{2 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^3}-\frac{2 B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^3}-\frac{(2 B g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{a f-b g}\right )}{x} \, dx,x,g+f x\right )}{f^3}+\frac{(2 B g n) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{c f-d g}\right )}{x} \, dx,x,g+f x\right )}{f^3}\\ &=\frac{A x}{f^2}-\frac{b B g^2 n \log (a+b x)}{f^3 (a f-b g)}+\frac{B (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b f^2}-\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{f^3 (g+f x)}-\frac{B (b c-a d) n \log (c+d x)}{b d f^2}+\frac{B d g^2 n \log (c+d x)}{f^3 (c f-d g)}+\frac{B (b c-a d) g^2 n \log (g+f x)}{f^2 (a f-b g) (c f-d g)}+\frac{2 B g n \log \left (\frac{f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^3}-\frac{2 g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^3}-\frac{2 B g n \log \left (\frac{f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^3}+\frac{2 B g n \text{Li}_2\left (-\frac{b (g+f x)}{a f-b g}\right )}{f^3}-\frac{2 B g n \text{Li}_2\left (-\frac{d (g+f x)}{c f-d g}\right )}{f^3}\\ \end{align*}
Mathematica [A] time = 0.593835, size = 295, normalized size = 0.92 \[ \frac{2 B g n \left (\text{PolyLog}\left (2,\frac{b (f x+g)}{b g-a f}\right )-\text{PolyLog}\left (2,\frac{d (f x+g)}{d g-c f}\right )+\log (f x+g) \left (\log \left (\frac{f (a+b x)}{a f-b g}\right )-\log \left (\frac{f (c+d x)}{c f-d g}\right )\right )\right )-\frac{g^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{f x+g}-2 g \log (f x+g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{B f (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}+\frac{B g^2 n (b \log (a+b x) (d g-c f)+d (a f-b g) \log (c+d x)+f (b c-a d) \log (f x+g))}{(a f-b g) (c f-d g)}-\frac{B f n (b c-a d) \log (c+d x)}{b d}+A f x}{f^3} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.084, size = 0, normalized size = 0. \begin{align*} \int{ \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) \left ( f+{\frac{g}{x}} \right ) ^{-2}}\, 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*} -A{\left (\frac{g^{2}}{f^{4} x + f^{3} g} - \frac{x}{f^{2}} + \frac{2 \, g \log \left (f x + g\right )}{f^{3}}\right )} - B \int -\frac{x^{2} \log \left ({\left (b x + a\right )}^{n}\right ) - x^{2} \log \left ({\left (d x + c\right )}^{n}\right ) + x^{2} \log \left (e\right )}{f^{2} x^{2} + 2 \, f g x + g^{2}}\,{d x} \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 x^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A x^{2}}{f^{2} x^{2} + 2 \, f g x + g^{2}}, 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}{{\left (f + \frac{g}{x}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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