Optimal. Leaf size=147 \[ \frac {\log (f+g x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g}-\frac {B n \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {B n \log (f+g x) \log \left (-\frac {g (a+b x)}{b f-a g}\right )}{g}+\frac {B n \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {B n \log (f+g x) \log \left (-\frac {g (c+d x)}{d f-c g}\right )}{g} \]
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Rubi [A] time = 0.23, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2524, 2418, 2394, 2393, 2391} \[ -\frac {B n \text {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B n \text {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {\log (f+g x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g}-\frac {B n \log (f+g x) \log \left (-\frac {g (a+b x)}{b f-a g}\right )}{g}+\frac {B n \log (f+g x) \log \left (-\frac {g (c+d x)}{d f-c g}\right )}{g} \]
Antiderivative was successfully verified.
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Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rubi steps
\begin {align*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f+g x} \, dx &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{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 (f+g x)}{a+b x} \, dx}{g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{g}-\frac {(B n) \int \left (\frac {b \log (f+g x)}{a+b x}-\frac {d \log (f+g x)}{c+d x}\right ) \, dx}{g}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{g}-\frac {(b B n) \int \frac {\log (f+g x)}{a+b x} \, dx}{g}+\frac {(B d n) \int \frac {\log (f+g x)}{c+d x} \, dx}{g}\\ &=-\frac {B n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{g}+\frac {B n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+(B n) \int \frac {\log \left (\frac {g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx-(B n) \int \frac {\log \left (\frac {g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx\\ &=-\frac {B n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{g}+\frac {B n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+\frac {(B n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b f+a g}\right )}{x} \, dx,x,f+g x\right )}{g}-\frac {(B n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d f+c g}\right )}{x} \, dx,x,f+g x\right )}{g}\\ &=-\frac {B n \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{g}+\frac {B n \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {B n \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B n \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 122, normalized size = 0.83 \[ \frac {\log (f+g x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-B n \log \left (\frac {g (a+b x)}{a g-b f}\right )+A+B n \log \left (\frac {g (c+d x)}{c g-d f}\right )\right )-B n \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )+B n \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}{g x + f}, x\right ) \]
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 = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{g x +f}\, dx \]
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 \[ -B \int -\frac {\log \left ({\left (b x + a\right )}^{n}\right ) - \log \left ({\left (d x + c\right )}^{n}\right ) + \log \relax (e)}{g x + f}\,{d x} + \frac {A \log \left (g x + f\right )}{g} \]
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 \[ \int \frac {A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{f+g\,x} \,d x \]
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 \[ \int \frac {A + B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{f + g x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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