Optimal. Leaf size=134 \[ \frac {g (b c-a d) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A+B n\right )}{d^2 i}+\frac {g (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d i}+\frac {B g n (b c-a d) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^2 i} \]
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Rubi [A] time = 0.39, antiderivative size = 223, normalized size of antiderivative = 1.66, number of steps used = 13, number of rules used = 10, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {2528, 2486, 31, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac {B g n (b c-a d) \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{d^2 i}-\frac {g (b c-a d) \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{d^2 i}-\frac {B g n (b c-a d) \log ^2(c+d x)}{2 d^2 i}-\frac {B g n (b c-a d) \log (c+d x)}{d^2 i}+\frac {B g n (b c-a d) \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{d^2 i}+\frac {B g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{d i}+\frac {A b g x}{d i} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2486
Rule 2524
Rule 2528
Rubi steps
\begin {align*} \int \frac {(a g+b g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{137 c+137 d x} \, dx &=\int \left (\frac {b g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{137 d}+\frac {(-b c+a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{137 d (c+d x)}\right ) \, dx\\ &=\frac {(b g) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{137 d}-\frac {((b c-a d) g) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{137 d}\\ &=\frac {A b g x}{137 d}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}+\frac {(b B g) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{137 d}+\frac {(B (b c-a d) g n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{137 d^2}\\ &=\frac {A b g x}{137 d}+\frac {B g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{137 d}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}+\frac {(B (b c-a d) g n) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{137 d^2}-\frac {(B (b c-a d) g n) \int \frac {1}{c+d x} \, dx}{137 d}\\ &=\frac {A b g x}{137 d}+\frac {B g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{137 d}-\frac {B (b c-a d) g n \log (c+d x)}{137 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}+\frac {(b B (b c-a d) g n) \int \frac {\log (c+d x)}{a+b x} \, dx}{137 d^2}-\frac {(B (b c-a d) g n) \int \frac {\log (c+d x)}{c+d x} \, dx}{137 d}\\ &=\frac {A b g x}{137 d}+\frac {B g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{137 d}-\frac {B (b c-a d) g n \log (c+d x)}{137 d^2}+\frac {B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{137 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}-\frac {(B (b c-a d) g n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{137 d^2}-\frac {(B (b c-a d) g n) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{137 d}\\ &=\frac {A b g x}{137 d}+\frac {B g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{137 d}-\frac {B (b c-a d) g n \log (c+d x)}{137 d^2}+\frac {B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{137 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}-\frac {B (b c-a d) g n \log ^2(c+d x)}{274 d^2}-\frac {(B (b c-a d) g n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{137 d^2}\\ &=\frac {A b g x}{137 d}+\frac {B g (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{137 d}-\frac {B (b c-a d) g n \log (c+d x)}{137 d^2}+\frac {B (b c-a d) g n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{137 d^2}-\frac {(b c-a d) g \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{137 d^2}-\frac {B (b c-a d) g n \log ^2(c+d x)}{274 d^2}+\frac {B (b c-a d) g n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{137 d^2}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 170, normalized size = 1.27 \[ \frac {g \left (-2 (b c-a d) \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+2 B d (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B n (b c-a d) \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B n (b c-a d) \log (c+d x)+2 A b d x\right )}{2 d^2 i} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A b g x + A a g + {\left (B b g x + B a g\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{d i x + c i}, x\right ) \]
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 = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {\left (b g x +a g \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{d i x +c i}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 4.05, size = 306, normalized size = 2.28 \[ A b g {\left (\frac {x}{d i} - \frac {c \log \left (d x + c\right )}{d^{2} i}\right )} + \frac {A a g \log \left (d i x + c i\right )}{d i} - \frac {{\left (b c g n - a d g n\right )} {\left (\log \left (b x + a\right ) \log \left (\frac {b d x + a d}{b c - a d} + 1\right ) + {\rm Li}_2\left (-\frac {b d x + a d}{b c - a d}\right )\right )} B}{d^{2} i} + \frac {{\left (a d g \log \relax (e) - {\left (g n + g \log \relax (e)\right )} b c\right )} B \log \left (d x + c\right )}{d^{2} i} + \frac {2 \, B a d g n \log \left (b x + a\right ) + 2 \, B b d g x \log \relax (e) + 2 \, {\left (b c g n - a d g n\right )} B \log \left (b x + a\right ) \log \left (d x + c\right ) - {\left (b c g n - a d g n\right )} B \log \left (d x + c\right )^{2} + 2 \, {\left (B b d g x - {\left (b c g - a d g\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \, {\left (B b d g x - {\left (b c g - a d g\right )} B \log \left (d x + c\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{2 \, d^{2} i} \]
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 \[ \int \frac {\left (a\,g+b\,g\,x\right )\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{c\,i+d\,i\,x} \,d x \]
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 \[ \frac {g \left (\int \frac {A a}{c + d x}\, dx + \int \frac {A b x}{c + d x}\, dx + \int \frac {B a \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx + \int \frac {B b x \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c + d x}\, dx\right )}{i} \]
Verification of antiderivative is not currently implemented for this CAS.
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