Optimal. Leaf size=150 \[ -\frac {d i \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2}-\frac {i (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b g^2 (a+b x)}+\frac {B d i n \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{b^2 g^2}-\frac {B i n (c+d x)}{b g^2 (a+b x)} \]
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Rubi [A] time = 0.38, antiderivative size = 233, normalized size of antiderivative = 1.55, number of steps used = 14, number of rules used = 11, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.268, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B d i n \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^2}+\frac {d i \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2}-\frac {i (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 g^2 (a+b x)}-\frac {B i n (b c-a d)}{b^2 g^2 (a+b x)}+\frac {B d i n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {B d i n \log ^2(a+b x)}{2 b^2 g^2}-\frac {B d i n \log (a+b x)}{b^2 g^2}+\frac {B d i n \log (c+d x)}{b^2 g^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 44
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {(113 c+113 d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^2} \, dx &=\int \left (\frac {113 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g^2 (a+b x)^2}+\frac {113 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b g^2 (a+b x)}\right ) \, dx\\ &=\frac {(113 d) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b g^2}+\frac {(113 (b c-a d)) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b g^2}\\ &=-\frac {113 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac {113 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {(113 B d n) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^2 g^2}+\frac {(113 B (b c-a d) n) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}\\ &=-\frac {113 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac {113 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {(113 B d n) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^2 g^2}+\frac {\left (113 B (b c-a d)^2 n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^2 g^2}\\ &=-\frac {113 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac {113 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}-\frac {(113 B d n) \int \frac {\log (a+b x)}{a+b x} \, dx}{b g^2}+\frac {\left (113 B d^2 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^2 g^2}+\frac {\left (113 B (b c-a d)^2 n\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^2 g^2}\\ &=-\frac {113 B (b c-a d) n}{b^2 g^2 (a+b x)}-\frac {113 B d n \log (a+b x)}{b^2 g^2}-\frac {113 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac {113 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}+\frac {113 B d n \log (c+d x)}{b^2 g^2}+\frac {113 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {(113 B d n) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^2 g^2}-\frac {(113 B d n) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b g^2}\\ &=-\frac {113 B (b c-a d) n}{b^2 g^2 (a+b x)}-\frac {113 B d n \log (a+b x)}{b^2 g^2}-\frac {113 B d n \log ^2(a+b x)}{2 b^2 g^2}-\frac {113 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac {113 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}+\frac {113 B d n \log (c+d x)}{b^2 g^2}+\frac {113 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}-\frac {(113 B d n) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^2 g^2}\\ &=-\frac {113 B (b c-a d) n}{b^2 g^2 (a+b x)}-\frac {113 B d n \log (a+b x)}{b^2 g^2}-\frac {113 B d n \log ^2(a+b x)}{2 b^2 g^2}-\frac {113 (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2 (a+b x)}+\frac {113 d \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2 g^2}+\frac {113 B d n \log (c+d x)}{b^2 g^2}+\frac {113 B d n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b^2 g^2}+\frac {113 B d n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b^2 g^2}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 189, normalized size = 1.26 \[ \frac {i \left (\frac {d \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2}-\frac {(b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b^2 (a+b x)}-\frac {B d n \left (-2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )-2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+\log ^2(a+b x)\right )}{2 b^2}-\frac {B n \left (\frac {b c-a d}{a+b x}+d \log (a+b x)-d \log (c+d x)\right )}{b^2}\right )}{g^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.98, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {A d i x + A c i + {\left (B d i x + B c i\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{b^{2} g^{2} x^{2} + 2 \, a b g^{2} x + a^{2} g^{2}}, x\right ) \]
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 = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right ) \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )}{\left (b g x +a g \right )^{2}}\, dx \]
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 \[ -B c i n {\left (\frac {1}{b^{2} g^{2} x + a b g^{2}} + \frac {d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac {d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} + B d i {\left (\frac {{\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - {\left ({\left (b x + a\right )} \log \left (b x + a\right ) + a\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{b^{3} g^{2} x + a b^{2} g^{2}} + \int \frac {b^{2} d x^{2} \log \relax (e) + b^{2} c x \log \relax (e) - a b c n + a^{2} d n - {\left (a b c n - a^{2} d n + {\left (b^{2} c n - a b d n\right )} x\right )} \log \left (b x + a\right )}{b^{4} d g^{2} x^{3} + a^{2} b^{2} c g^{2} + {\left (b^{4} c g^{2} + 2 \, a b^{3} d g^{2}\right )} x^{2} + {\left (2 \, a b^{3} c g^{2} + a^{2} b^{2} d g^{2}\right )} x}\,{d x}\right )} + A d i {\left (\frac {a}{b^{3} g^{2} x + a b^{2} g^{2}} + \frac {\log \left (b x + a\right )}{b^{2} g^{2}}\right )} - \frac {B c i \log \left (e {\left (\frac {b x}{d x + c} + \frac {a}{d x + c}\right )}^{n}\right )}{b^{2} g^{2} x + a b g^{2}} - \frac {A c i}{b^{2} g^{2} x + a b g^{2}} \]
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 \[ \int \frac {\left (c\,i+d\,i\,x\right )\,\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}{{\left (a\,g+b\,g\,x\right )}^2} \,d x \]
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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