Optimal. Leaf size=143 \[ g \log (x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {B f (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}-\frac {B f n (b c-a d) \log (c+d x)}{b d}-B g n \text {Li}_2\left (-\frac {b x}{a}\right )-B g n \log (x) \log \left (\frac {b x}{a}+1\right )+A f x+B g n \text {Li}_2\left (-\frac {d x}{c}\right )+B g n \log (x) \log \left (\frac {d x}{c}+1\right ) \]
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Rubi [A] time = 0.22, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2528, 2486, 31, 2524, 2357, 2317, 2391} \[ -B g n \text {PolyLog}\left (2,-\frac {b x}{a}\right )+B g n \text {PolyLog}\left (2,-\frac {d x}{c}\right )+g \log (x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {B f (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}-\frac {B f n (b c-a d) \log (c+d x)}{b d}-B g n \log (x) \log \left (\frac {b x}{a}+1\right )+A f x+B g n \log (x) \log \left (\frac {d x}{c}+1\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 2317
Rule 2357
Rule 2391
Rule 2486
Rule 2524
Rule 2528
Rubi steps
\begin {align*} \int \left (f+\frac {g}{x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx &=\int \left (f \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{x}\right ) \, dx\\ &=f \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx+g \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{x} \, dx\\ &=A f x+g \log (x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+(B f) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx-(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 (x)}{a+b x} \, dx\\ &=A f x+\frac {B f (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+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 n) \int \frac {1}{c+d x} \, dx}{b}-(B g n) \int \left (\frac {b \log (x)}{a+b x}-\frac {d \log (x)}{c+d x}\right ) \, dx\\ &=A f x+\frac {B f (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+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 n \log (c+d x)}{b d}-(b B g n) \int \frac {\log (x)}{a+b x} \, dx+(B d g n) \int \frac {\log (x)}{c+d x} \, dx\\ &=A f x-B g n \log (x) \log \left (1+\frac {b x}{a}\right )+\frac {B f (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+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 n \log (c+d x)}{b d}+B g n \log (x) \log \left (1+\frac {d x}{c}\right )+(B g n) \int \frac {\log \left (1+\frac {b x}{a}\right )}{x} \, dx-(B g n) \int \frac {\log \left (1+\frac {d x}{c}\right )}{x} \, dx\\ &=A f x-B g n \log (x) \log \left (1+\frac {b x}{a}\right )+\frac {B f (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+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 n \log (c+d x)}{b d}+B g n \log (x) \log \left (1+\frac {d x}{c}\right )-B g n \text {Li}_2\left (-\frac {b x}{a}\right )+B g n \text {Li}_2\left (-\frac {d x}{c}\right )\\ \end {align*}
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Mathematica [A] time = 0.10, size = 135, normalized size = 0.94 \[ g \log (x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {B f (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}-\frac {B f n (b c-a d) \log (c+d x)}{b d}-B g n \left (\log (x) \left (\log \left (\frac {b x}{a}+1\right )-\log \left (\frac {d x}{c}+1\right )\right )+\text {Li}_2\left (-\frac {b x}{a}\right )-\text {Li}_2\left (-\frac {d x}{c}\right )\right )+A f x \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A f x + A g + {\left (B f x + B g\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[ \int \left (f +\frac {g}{x}\right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ B f n {\left (\frac {a \log \left (b x + a\right )}{b} - \frac {c \log \left (d x + c\right )}{d}\right )} + B f x \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right ) + A f x - B g \int -\frac {\log \left ({\left (b x + a\right )}^{n}\right ) - \log \left ({\left (d x + c\right )}^{n}\right ) + \log \relax (e)}{x}\,{d x} + A g \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (f+\frac {g}{x}\right )\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}\right ) \left (f x + g\right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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