Optimal. Leaf size=80 \[ -\frac {\log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d i}-\frac {B n \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d i} \]
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Rubi [A] time = 0.20, antiderivative size = 128, normalized size of antiderivative = 1.60, number of steps used = 9, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {2524, 2418, 2394, 2393, 2391, 2390, 12, 2301} \[ -\frac {B n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d i}+\frac {\log (c i+d i x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d i}-\frac {B n \log (c i+d i x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d i}+\frac {B n \log ^2(i (c+d x))}{2 d i} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2301
Rule 2390
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 )}{138 c+138 d x} \, dx &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}-\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 (138 c+138 d x)}{a+b x} \, dx}{138 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}-\frac {(B n) \int \left (\frac {b \log (138 c+138 d x)}{a+b x}-\frac {d \log (138 c+138 d x)}{c+d x}\right ) \, dx}{138 d}\\ &=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}+\frac {1}{138} (B n) \int \frac {\log (138 c+138 d x)}{c+d x} \, dx-\frac {(b B n) \int \frac {\log (138 c+138 d x)}{a+b x} \, dx}{138 d}\\ &=-\frac {B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (138 c+138 d x)}{138 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}+(B n) \int \frac {\log \left (\frac {138 d (a+b x)}{-138 b c+138 a d}\right )}{138 c+138 d x} \, dx+\frac {(B n) \operatorname {Subst}\left (\int \frac {138 \log (x)}{x} \, dx,x,138 c+138 d x\right )}{19044 d}\\ &=-\frac {B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (138 c+138 d x)}{138 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}+\frac {(B n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,138 c+138 d x\right )}{138 d}+\frac {(B n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-138 b c+138 a d}\right )}{x} \, dx,x,138 c+138 d x\right )}{138 d}\\ &=\frac {B n \log ^2(138 (c+d x))}{276 d}-\frac {B n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (138 c+138 d x)}{138 d}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (138 c+138 d x)}{138 d}-\frac {B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{138 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 101, normalized size = 1.26 \[ \frac {\log (i (c+d x)) \left (2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-2 B n \log \left (\frac {d (a+b x)}{a d-b c}\right )+2 A+B n \log (i (c+d x))\right )-2 B n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2 d i} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A}{d i x + c i}, x\right ) \]
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 = 0.37, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{d i x +c i}\, dx \]
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 \[ -\frac {1}{2} \, B {\left (\frac {2 \, n \log \left (b x + a\right ) \log \left (d x + c\right ) - n \log \left (d x + c\right )^{2} - 2 \, \log \left (d x + c\right ) \log \left ({\left (b x + a\right )}^{n}\right ) + 2 \, \log \left (d x + c\right ) \log \left ({\left (d x + c\right )}^{n}\right )}{d i} - 2 \, \int \frac {n \log \left (b x + a\right ) + \log \relax (e)}{d i x + c i}\,{d x}\right )} + \frac {A \log \left (d i x + c i\right )}{d i} \]
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 \[ \int \frac {A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{c\,i+d\,i\,x} \,d x \]
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 \[ \frac {\int \frac {A}{c + d x}\, dx + \int \frac {B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx}{i} \]
Verification of antiderivative is not currently implemented for this CAS.
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