Optimal. Leaf size=217 \[ -\frac {g \log (f x+g) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{f^2}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}-\frac {B n (b c-a d) \log (c+d x)}{b d f}+\frac {B g n \text {Li}_2\left (-\frac {b (g+f x)}{a f-b g}\right )}{f^2}+\frac {B g n \log (f x+g) \log \left (\frac {f (a+b x)}{a f-b g}\right )}{f^2}+\frac {A x}{f}-\frac {B g n \text {Li}_2\left (-\frac {d (g+f x)}{c f-d g}\right )}{f^2}-\frac {B g n \log (f x+g) \log \left (\frac {f (c+d x)}{c f-d g}\right )}{f^2} \]
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Rubi [A] time = 0.33, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2528, 2486, 31, 2524, 2418, 2394, 2393, 2391} \[ \frac {B g n \text {PolyLog}\left (2,-\frac {b (f x+g)}{a f-b g}\right )}{f^2}-\frac {B g n \text {PolyLog}\left (2,-\frac {d (f x+g)}{c f-d g}\right )}{f^2}-\frac {g \log (f x+g) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{f^2}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}-\frac {B n (b c-a d) \log (c+d x)}{b d f}+\frac {B g n \log (f x+g) \log \left (\frac {f (a+b x)}{a f-b g}\right )}{f^2}+\frac {A x}{f}-\frac {B g n \log (f x+g) \log \left (\frac {f (c+d x)}{c f-d g}\right )}{f^2} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2486
Rule 2524
Rule 2528
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+\frac {g}{x}} \, dx &=\int \left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{f (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}-\frac {g \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{g+f x} \, dx}{f}\\ &=\frac {A x}{f}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}+\frac {B \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{f}+\frac {(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^2}\\ &=\frac {A x}{f}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac {(B (b c-a d) n) \int \frac {1}{c+d x} \, dx}{b f}+\frac {(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^2}\\ &=\frac {A x}{f}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}-\frac {B (b c-a d) n \log (c+d x)}{b d f}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}+\frac {(b B g n) \int \frac {\log (g+f x)}{a+b x} \, dx}{f^2}-\frac {(B d g n) \int \frac {\log (g+f x)}{c+d x} \, dx}{f^2}\\ &=\frac {A x}{f}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}-\frac {B (b c-a d) n \log (c+d x)}{b d f}+\frac {B g n \log \left (\frac {f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^2}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac {B g n \log \left (\frac {f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^2}-\frac {(B g n) \int \frac {\log \left (\frac {f (a+b x)}{a f-b g}\right )}{g+f x} \, dx}{f}+\frac {(B g n) \int \frac {\log \left (\frac {f (c+d x)}{c f-d g}\right )}{g+f x} \, dx}{f}\\ &=\frac {A x}{f}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}-\frac {B (b c-a d) n \log (c+d x)}{b d f}+\frac {B g n \log \left (\frac {f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^2}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac {B g n \log \left (\frac {f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^2}-\frac {(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^2}+\frac {(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^2}\\ &=\frac {A x}{f}+\frac {B (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b f}-\frac {B (b c-a d) n \log (c+d x)}{b d f}+\frac {B g n \log \left (\frac {f (a+b x)}{a f-b g}\right ) \log (g+f x)}{f^2}-\frac {g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (g+f x)}{f^2}-\frac {B g n \log \left (\frac {f (c+d x)}{c f-d g}\right ) \log (g+f x)}{f^2}+\frac {B g n \text {Li}_2\left (-\frac {b (g+f x)}{a f-b g}\right )}{f^2}-\frac {B g n \text {Li}_2\left (-\frac {d (g+f x)}{c f-d g}\right )}{f^2}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 185, normalized size = 0.85 \[ \frac {-g \log (f x+g) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+\frac {B f (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b}+B g n \left (\log (f x+g) \left (\log \left (\frac {f (a+b x)}{a f-b g}\right )-\log \left (\frac {f (c+d x)}{c f-d g}\right )\right )+\text {Li}_2\left (\frac {b (g+f x)}{b g-a f}\right )-\text {Li}_2\left (\frac {d (g+f x)}{d g-c f}\right )\right )-\frac {B f n (b c-a d) \log (c+d x)}{b d}+A f x}{f^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B x \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A x}{f x + g}, x\right ) \]
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 = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{f +\frac {g}{x}}\, dx \]
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 \[ A {\left (\frac {x}{f} - \frac {g \log \left (f x + g\right )}{f^{2}}\right )} - B \int -\frac {x \log \left ({\left (b x + a\right )}^{n}\right ) - x \log \left ({\left (d x + c\right )}^{n}\right ) + x \log \relax (e)}{f x + g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{f+\frac {g}{x}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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