Optimal. Leaf size=263 \[ -2 B f g n \text{PolyLog}\left (2,-\frac{b x}{a}\right )+2 B f g n \text{PolyLog}\left (2,-\frac{d x}{c}\right )+2 f g \log (x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{g^2 (a+b x) (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{a (c+d x) \left (a-\frac{c (a+b x)}{c+d x}\right )}+\frac{B f^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{B f^2 n (b c-a d) \log (c+d x)}{b d}+\frac{B g^2 n (b c-a d) \log \left (a-\frac{c (a+b x)}{c+d x}\right )}{a c}-2 B f g n \log (x) \log \left (\frac{b x}{a}+1\right )+A f^2 x+2 B f g n \log (x) \log \left (\frac{d x}{c}+1\right ) \]
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Rubi [A] time = 0.337362, antiderivative size = 242, normalized size of antiderivative = 0.92, number of steps used = 16, number of rules used = 10, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {2528, 2486, 31, 2525, 12, 72, 2524, 2357, 2317, 2391} \[ -2 B f g n \text{PolyLog}\left (2,-\frac{b x}{a}\right )+2 B f g n \text{PolyLog}\left (2,-\frac{d x}{c}\right )+2 f g \log (x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{g^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{x}+\frac{B f^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{B f^2 n (b c-a d) \log (c+d x)}{b d}+\frac{B g^2 n \log (x) (b c-a d)}{a c}-2 B f g n \log (x) \log \left (\frac{b x}{a}+1\right )-\frac{b B g^2 n \log (a+b x)}{a}+A f^2 x+2 B f g n \log (x) \log \left (\frac{d x}{c}+1\right )+\frac{B d g^2 n \log (c+d x)}{c} \]
Antiderivative was successfully verified.
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Rule 2528
Rule 2486
Rule 31
Rule 2525
Rule 12
Rule 72
Rule 2524
Rule 2357
Rule 2317
Rule 2391
Rubi steps
\begin{align*} \int \left (f+\frac{g}{x}\right )^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (f^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x^2}+\frac{2 f g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}\right ) \, dx\\ &=f^2 \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx+(2 f g) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{x} \, dx+g^2 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{x^2} \, dx\\ &=A f^2 x-\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}+2 f g \log (x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )+\left (B f^2\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx-(2 B f g n) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (x)}{a+b x} \, dx+\left (B g^2 n\right ) \int \frac{b c-a d}{x (a+b x) (c+d x)} \, dx\\ &=A f^2 x+\frac{B f^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}+2 f g \log (x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-\frac{\left (B (b c-a d) f^2 n\right ) \int \frac{1}{c+d x} \, dx}{b}-(2 B f g n) \int \left (\frac{b \log (x)}{a+b x}-\frac{d \log (x)}{c+d x}\right ) \, dx+\left (B (b c-a d) g^2 n\right ) \int \frac{1}{x (a+b x) (c+d x)} \, dx\\ &=A f^2 x+\frac{B f^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}+2 f g \log (x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-\frac{B (b c-a d) f^2 n \log (c+d x)}{b d}-(2 b B f g n) \int \frac{\log (x)}{a+b x} \, dx+(2 B d f g n) \int \frac{\log (x)}{c+d x} \, dx+\left (B (b c-a d) g^2 n\right ) \int \left (\frac{1}{a c x}+\frac{b^2}{a (-b c+a d) (a+b x)}+\frac{d^2}{c (b c-a d) (c+d x)}\right ) \, dx\\ &=A f^2 x+\frac{B (b c-a d) g^2 n \log (x)}{a c}-\frac{b B g^2 n \log (a+b x)}{a}-2 B f g n \log (x) \log \left (1+\frac{b x}{a}\right )+\frac{B f^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}+2 f g \log (x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-\frac{B (b c-a d) f^2 n \log (c+d x)}{b d}+\frac{B d g^2 n \log (c+d x)}{c}+2 B f g n \log (x) \log \left (1+\frac{d x}{c}\right )+(2 B f g n) \int \frac{\log \left (1+\frac{b x}{a}\right )}{x} \, dx-(2 B f g n) \int \frac{\log \left (1+\frac{d x}{c}\right )}{x} \, dx\\ &=A f^2 x+\frac{B (b c-a d) g^2 n \log (x)}{a c}-\frac{b B g^2 n \log (a+b x)}{a}-2 B f g n \log (x) \log \left (1+\frac{b x}{a}\right )+\frac{B f^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{g^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{x}+2 f g \log (x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )-\frac{B (b c-a d) f^2 n \log (c+d x)}{b d}+\frac{B d g^2 n \log (c+d x)}{c}+2 B f g n \log (x) \log \left (1+\frac{d x}{c}\right )-2 B f g n \text{Li}_2\left (-\frac{b x}{a}\right )+2 B f g n \text{Li}_2\left (-\frac{d x}{c}\right )\\ \end{align*}
Mathematica [A] time = 0.230476, size = 217, normalized size = 0.83 \[ -2 B f g n \left (\text{PolyLog}\left (2,-\frac{b x}{a}\right )-\text{PolyLog}\left (2,-\frac{d x}{c}\right )+\log (x) \left (\log \left (\frac{b x}{a}+1\right )-\log \left (\frac{d x}{c}+1\right )\right )\right )+2 f g \log (x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-\frac{g^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{x}+\frac{B f^2 (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}-\frac{B f^2 n (b c-a d) \log (c+d x)}{b d}+\frac{B g^2 n (\log (x) (b c-a d)-b c \log (a+b x)+a d \log (c+d x))}{a c}+A f^2 x \]
Antiderivative was successfully verified.
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Maple [F] time = 0.086, size = 0, normalized size = 0. \begin{align*} \int \left ( f+{\frac{g}{x}} \right ) ^{2} \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*} B f^{2} n{\left (\frac{a \log \left (b x + a\right )}{b} - \frac{c \log \left (d x + c\right )}{d}\right )} - B g^{2} n{\left (\frac{b \log \left (b x + a\right )}{a} - \frac{d \log \left (d x + c\right )}{c} - \frac{{\left (b c - a d\right )} \log \left (x\right )}{a c}\right )} + B f^{2} x \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) + A f^{2} x - 2 \, B f g \int -\frac{\log \left ({\left (b x + a\right )}^{n}\right ) - \log \left ({\left (d x + c\right )}^{n}\right ) + \log \left (e\right )}{x}\,{d x} + 2 \, A f g \log \left (x\right ) - \frac{B g^{2} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{x} - \frac{A g^{2}}{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{A f^{2} x^{2} + 2 \, A f g x + A g^{2} +{\left (B f^{2} x^{2} + 2 \, B f g x + B g^{2}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{x^{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{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}{\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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