Optimal. Leaf size=83 \[ -\frac {\log \left (-\frac {b c-a d}{d (a+b x)}\right ) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{b g}-\frac {2 B \text {Li}_2\left (\frac {b c-a d}{d (a+b x)}+1\right )}{b g} \]
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Rubi [A] time = 0.29, antiderivative size = 121, normalized size of antiderivative = 1.46, number of steps used = 10, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2524, 12, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac {2 B \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{b g}+\frac {\log (a g+b g x) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{b g}-\frac {2 B \log (a g+b g x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b g}+\frac {B \log ^2(g (a+b x))}{b g} \]
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 (\frac {e (c+d x)^2}{(a+b x)^2}\right )}{a g+b g x} \, dx &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}-\frac {B \int \frac {(a+b x)^2 \left (\frac {2 d e (c+d x)}{(a+b x)^2}-\frac {2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (a g+b g x)}{e (c+d x)^2} \, dx}{b g}\\ &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}-\frac {B \int \frac {(a+b x)^2 \left (\frac {2 d e (c+d x)}{(a+b x)^2}-\frac {2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (a g+b g x)}{(c+d x)^2} \, dx}{b e g}\\ &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}-\frac {B \int \left (-\frac {2 b e \log (a g+b g x)}{a+b x}+\frac {2 d e \log (a g+b g x)}{c+d x}\right ) \, dx}{b e g}\\ &=\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac {(2 B) \int \frac {\log (a g+b g x)}{a+b x} \, dx}{g}-\frac {(2 B d) \int \frac {\log (a g+b g x)}{c+d x} \, dx}{b g}\\ &=-\frac {2 B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+(2 B) \int \frac {\log \left (\frac {b g (c+d x)}{b c g-a d g}\right )}{a g+b g x} \, dx+\frac {(2 B) \operatorname {Subst}\left (\int \frac {g \log (x)}{x} \, dx,x,a g+b g x\right )}{b g^2}\\ &=-\frac {2 B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}+\frac {(2 B) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a g+b g x\right )}{b g}+\frac {(2 B) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c g-a d g}\right )}{x} \, dx,x,a g+b g x\right )}{b g}\\ &=\frac {B \log ^2(g (a+b x))}{b g}-\frac {2 B \log \left (\frac {b (c+d x)}{b c-a d}\right ) \log (a g+b g x)}{b g}+\frac {\left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log (a g+b g x)}{b g}-\frac {2 B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b g}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 87, normalized size = 1.05 \[ \frac {\log (a+b x) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )-2 B \log \left (\frac {b (c+d x)}{b c-a d}\right )+B \log (a+b x)+A\right )-2 B \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )}{b g} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {B \log \left (\frac {d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ) + A}{b g x + a g}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A}{b g x + a g}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 265, normalized size = 3.19 \[ \frac {2 B a d \ln \left (\frac {1}{b x +a}\right ) \ln \left (-\frac {-d +\frac {a d -b c}{b x +a}}{d}\right )}{\left (a d -b c \right ) b g}-\frac {2 B c \ln \left (\frac {1}{b x +a}\right ) \ln \left (-\frac {-d +\frac {a d -b c}{b x +a}}{d}\right )}{\left (a d -b c \right ) g}+\frac {2 B a d \dilog \left (-\frac {-d +\frac {a d -b c}{b x +a}}{d}\right )}{\left (a d -b c \right ) b g}-\frac {2 B c \dilog \left (-\frac {-d +\frac {a d -b c}{b x +a}}{d}\right )}{\left (a d -b c \right ) g}-\frac {B \ln \left (\frac {1}{b x +a}\right ) \ln \left (\frac {\left (\frac {a d}{b x +a}-\frac {b c}{b x +a}-d \right )^{2} e}{b^{2}}\right )}{b g}-\frac {A \ln \left (\frac {1}{b x +a}\right )}{b g} \]
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 \[ B {\left (\frac {2 \, \log \left (b x + a\right ) \log \left (d x + c\right )}{b g} - \int -\frac {b d x \log \relax (e) + b c \log \relax (e) - 2 \, {\left (2 \, b d x + b c + a d\right )} \log \left (b x + a\right )}{b^{2} d g x^{2} + a b c g + {\left (b^{2} c g + a b d g\right )} x}\,{d x}\right )} + \frac {A \log \left (b g x + a g\right )}{b g} \]
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 \[ \int \frac {A+B\,\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )}{a\,g+b\,g\,x} \,d x \]
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 \[ \frac {\int \frac {A}{a + b x}\, dx + \int \frac {B \log {\left (\frac {c^{2} e}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {2 c d e x}{a^{2} + 2 a b x + b^{2} x^{2}} + \frac {d^{2} e x^{2}}{a^{2} + 2 a b x + b^{2} x^{2}} \right )}}{a + b x}\, dx}{g} \]
Verification of antiderivative is not currently implemented for this CAS.
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