Optimal. Leaf size=44 \[ \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 B g i (b c-a d)} \]
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Rubi [C] time = 0.58, antiderivative size = 304, normalized size of antiderivative = 6.91, number of steps used = 20, number of rules used = 9, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.225, Rules used = {2528, 2524, 12, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g i (b c-a d)}+\frac {B \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g i (b c-a d)}+\frac {\log (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i (b c-a d)}-\frac {\log (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i (b c-a d)}-\frac {B \log ^2(a+b x)}{2 g i (b c-a d)}-\frac {B \log ^2(c+d x)}{2 g i (b c-a d)}+\frac {B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g i (b c-a d)}+\frac {B \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g i (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2301
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2528
Rubi steps
\begin {align*} \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(35 c+35 d x) (a g+b g x)} \, dx &=\int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{35 (b c-a d) g (a+b x)}-\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{35 (b c-a d) g (c+d x)}\right ) \, dx\\ &=\frac {b \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{35 (b c-a d) g}-\frac {d \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{35 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{35 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{35 (b c-a d) g}-\frac {B \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{35 (b c-a d) g}+\frac {B \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{35 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{35 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{35 (b c-a d) g}-\frac {B \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{35 (b c-a d) e g}+\frac {B \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{35 (b c-a d) e g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{35 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{35 (b c-a d) g}-\frac {B \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{35 (b c-a d) e g}+\frac {B \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{35 (b c-a d) e g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{35 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{35 (b c-a d) g}-\frac {(b B) \int \frac {\log (a+b x)}{a+b x} \, dx}{35 (b c-a d) g}+\frac {(b B) \int \frac {\log (c+d x)}{a+b x} \, dx}{35 (b c-a d) g}+\frac {(B d) \int \frac {\log (a+b x)}{c+d x} \, dx}{35 (b c-a d) g}-\frac {(B d) \int \frac {\log (c+d x)}{c+d x} \, dx}{35 (b c-a d) g}\\ &=\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{35 (b c-a d) g}+\frac {B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{35 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{35 (b c-a d) g}+\frac {B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{35 (b c-a d) g}-\frac {B \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{35 (b c-a d) g}-\frac {B \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{35 (b c-a d) g}-\frac {(b B) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{35 (b c-a d) g}-\frac {(B d) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{35 (b c-a d) g}\\ &=-\frac {B \log ^2(a+b x)}{70 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{35 (b c-a d) g}+\frac {B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{35 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{35 (b c-a d) g}-\frac {B \log ^2(c+d x)}{70 (b c-a d) g}+\frac {B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{35 (b c-a d) g}-\frac {B \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{35 (b c-a d) g}-\frac {B \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{35 (b c-a d) g}\\ &=-\frac {B \log ^2(a+b x)}{70 (b c-a d) g}+\frac {\log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{35 (b c-a d) g}+\frac {B \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{35 (b c-a d) g}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{35 (b c-a d) g}-\frac {B \log ^2(c+d x)}{70 (b c-a d) g}+\frac {B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{35 (b c-a d) g}+\frac {B \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{35 (b c-a d) g}+\frac {B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{35 (b c-a d) g}\\ \end {align*}
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Mathematica [C] time = 0.12, size = 207, normalized size = 4.70 \[ \frac {2 A \log (a+b x)+2 B \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-2 B \log (c+d x) \log \left (\frac {e (a+b x)}{c+d x}\right )+2 B \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )+2 B \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+2 B \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+2 B \log (c+d x) \log \left (\frac {d (a+b x)}{a d-b c}\right )-B \log ^2(a+b x)-2 A \log (c+d x)-B \log ^2(c+d x)}{2 g i (b c-a d)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 60, normalized size = 1.36 \[ \frac {B \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, A \log \left (\frac {b e x + a e}{d x + c}\right )}{2 \, {\left (b c - a d\right )} g i} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.35, size = 113, normalized size = 2.57 \[ -\frac {{\left (B i e \log \left (\frac {b x e + a e}{d x + c}\right )^{2} + 2 \, A i e \log \left (\frac {b x e + a e}{d x + c}\right )\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{2 \, g} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 201, normalized size = 4.57 \[ -\frac {B a d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right )^{2} g i}+\frac {B b c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )^{2}}{2 \left (a d -b c \right )^{2} g i}-\frac {A a d \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} g i}+\frac {A b c \ln \left (\frac {b e}{d}+\frac {\left (a d -b c \right ) e}{\left (d x +c \right ) d}\right )}{\left (a d -b c \right )^{2} g i} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.14, size = 172, normalized size = 3.91 \[ B {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + A {\left (\frac {\log \left (b x + a\right )}{{\left (b c - a d\right )} g i} - \frac {\log \left (d x + c\right )}{{\left (b c - a d\right )} g i}\right )} - \frac {{\left (\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (d x + c\right ) + \log \left (d x + c\right )^{2}\right )} B}{2 \, {\left (b c g i - a d g i\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.74, size = 69, normalized size = 1.57 \[ -\frac {B\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2-A\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,4{}\mathrm {i}}{2\,g\,i\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.13, size = 170, normalized size = 3.86 \[ A \left (\frac {\log {\left (x + \frac {- \frac {a^{2} d^{2}}{a d - b c} + \frac {2 a b c d}{a d - b c} + a d - \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{g i \left (a d - b c\right )} - \frac {\log {\left (x + \frac {\frac {a^{2} d^{2}}{a d - b c} - \frac {2 a b c d}{a d - b c} + a d + \frac {b^{2} c^{2}}{a d - b c} + b c}{2 b d} \right )}}{g i \left (a d - b c\right )}\right ) - \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a d g i - 2 b c g i} \]
Verification of antiderivative is not currently implemented for this CAS.
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