Optimal. Leaf size=130 \[ -\frac{4 B (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^2 (a+b x) (b c-a d)}-\frac{(c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^2 (a+b x) (b c-a d)}-\frac{8 B^2 (c+d x)}{g^2 (a+b x) (b c-a d)} \]
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Rubi [C] time = 0.888483, antiderivative size = 480, normalized size of antiderivative = 3.69, number of steps used = 26, number of rules used = 11, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.324, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{8 B^2 d \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac{8 B^2 d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac{4 B d \log (a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^2 (b c-a d)}-\frac{4 B \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^2 (a+b x)}+\frac{4 B d \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^2 (b c-a d)}-\frac{\left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{b g^2 (a+b x)}+\frac{4 B^2 d \log ^2(a+b x)}{b g^2 (b c-a d)}+\frac{4 B^2 d \log ^2(c+d x)}{b g^2 (b c-a d)}-\frac{8 B^2 d \log (a+b x)}{b g^2 (b c-a d)}-\frac{8 B^2 d \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}+\frac{8 B^2 d \log (c+d x)}{b g^2 (b c-a d)}-\frac{8 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac{8 B^2}{b g^2 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 12
Rule 2528
Rule 44
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{(2 B) \int \frac{2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (a+b x)^2 (c+d x)} \, dx}{b g}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{(4 B (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^2 (c+d x)} \, dx}{b g^2}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{(4 B (b c-a d)) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) (a+b x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b g^2}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{(4 B) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^2} \, dx}{g^2}-\frac{(4 B d) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx}{(b c-a d) g^2}+\frac{\left (4 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{\left (4 B^2\right ) \int \frac{2 (b c-a d)}{(a+b x)^2 (c+d x)} \, dx}{b g^2}+\frac{\left (4 B^2 d\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{e (a+b x)^2} \, dx}{b (b c-a d) g^2}-\frac{\left (4 B^2 d\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{e (a+b x)^2} \, dx}{b (b c-a d) g^2}\\ &=-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{\left (8 B^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b g^2}+\frac{\left (4 B^2 d\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{(a+b x)^2} \, dx}{b (b c-a d) e g^2}-\frac{\left (4 B^2 d\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{(a+b x)^2} \, dx}{b (b c-a d) e g^2}\\ &=-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{\left (8 B^2 (b c-a d)\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 g^2}+\frac{\left (4 B^2 d\right ) \int \left (\frac{2 b e \log (a+b x)}{a+b x}-\frac{2 d e \log (a+b x)}{c+d x}\right ) \, dx}{b (b c-a d) e g^2}-\frac{\left (4 B^2 d\right ) \int \left (\frac{2 b e \log (c+d x)}{a+b x}-\frac{2 d e \log (c+d x)}{c+d x}\right ) \, dx}{b (b c-a d) e g^2}\\ &=-\frac{8 B^2}{b g^2 (a+b x)}-\frac{8 B^2 d \log (a+b x)}{b (b c-a d) g^2}-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{8 B^2 d \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{(b c-a d) g^2}-\frac{\left (8 B^2 d\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{(b c-a d) g^2}-\frac{\left (8 B^2 d^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b (b c-a d) g^2}+\frac{\left (8 B^2 d^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=-\frac{8 B^2}{b g^2 (a+b x)}-\frac{8 B^2 d \log (a+b x)}{b (b c-a d) g^2}-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{8 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{(b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b (b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b (b c-a d) g^2}+\frac{\left (8 B^2 d^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=-\frac{8 B^2}{b g^2 (a+b x)}-\frac{8 B^2 d \log (a+b x)}{b (b c-a d) g^2}+\frac{4 B^2 d \log ^2(a+b x)}{b (b c-a d) g^2}-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{8 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B^2 d \log ^2(c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b (b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b (b c-a d) g^2}\\ &=-\frac{8 B^2}{b g^2 (a+b x)}-\frac{8 B^2 d \log (a+b x)}{b (b c-a d) g^2}+\frac{4 B^2 d \log ^2(a+b x)}{b (b c-a d) g^2}-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{8 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B^2 d \log ^2(c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}-\frac{8 B^2 d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d) g^2}-\frac{8 B^2 d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}\\ \end{align*}
Mathematica [C] time = 0.450889, size = 321, normalized size = 2.47 \[ -\frac{\frac{4 B \left (-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+B d (a+b x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+(b c-a d) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+d (a+b x) \log (a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )-d (a+b x) \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+2 B (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )}{b c-a d}+\left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{b g^2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.09, size = 357, normalized size = 2.8 \begin{align*}{\frac{d{A}^{2}}{{g}^{2} \left ( ad-bc \right ) } \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}-8\,{\frac{d{B}^{2}}{b{g}^{2} \left ( dx+c \right ) } \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}+4\,{\frac{d{B}^{2}}{{g}^{2} \left ( ad-bc \right ) }\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}+{\frac{d{B}^{2}}{{g}^{2} \left ( ad-bc \right ) } \left ( \ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) \right ) ^{2} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}-4\,{\frac{dAB}{b{g}^{2} \left ( dx+c \right ) } \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}+2\,{\frac{dAB}{{g}^{2} \left ( ad-bc \right ) }\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.4819, size = 775, normalized size = 5.96 \begin{align*} -4 \,{\left ({\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 )} \log \left (\frac{b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac{{\left (b d x + a d\right )} \log \left (b x + a\right )^{2} +{\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \,{\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \,{\left (b d x + a d -{\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c g^{2} - a^{2} b d g^{2} +{\left (b^{3} c g^{2} - a b^{2} d g^{2}\right )} x}\right )} B^{2} - 2 \, A B{\left (\frac{\log \left (\frac{b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac{2}{b^{2} g^{2} x + a b g^{2}} + \frac{2 \, d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac{2 \, d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac{B^{2} \log \left (\frac{b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2}}{b^{2} g^{2} x + a b g^{2}} - \frac{A^{2}}{b^{2} g^{2} x + a b g^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.998379, size = 416, normalized size = 3.2 \begin{align*} -\frac{{\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} b c -{\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} a d +{\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \,{\left ({\left (A B + 2 \, B^{2}\right )} b d x +{\left (A B + 2 \, B^{2}\right )} b c\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x +{\left (a b^{2} c - a^{2} b d\right )} g^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 3.79672, size = 452, normalized size = 3.48 \begin{align*} - \frac{4 B d \left (A + 2 B\right ) \log{\left (x + \frac{4 A B a d^{2} + 4 A B b c d + 8 B^{2} a d^{2} + 8 B^{2} b c d - \frac{4 B a^{2} d^{3} \left (A + 2 B\right )}{a d - b c} + \frac{8 B a b c d^{2} \left (A + 2 B\right )}{a d - b c} - \frac{4 B b^{2} c^{2} d \left (A + 2 B\right )}{a d - b c}}{8 A B b d^{2} + 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac{4 B d \left (A + 2 B\right ) \log{\left (x + \frac{4 A B a d^{2} + 4 A B b c d + 8 B^{2} a d^{2} + 8 B^{2} b c d + \frac{4 B a^{2} d^{3} \left (A + 2 B\right )}{a d - b c} - \frac{8 B a b c d^{2} \left (A + 2 B\right )}{a d - b c} + \frac{4 B b^{2} c^{2} d \left (A + 2 B\right )}{a d - b c}}{8 A B b d^{2} + 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac{\left (- 2 A B - 4 B^{2}\right ) \log{\left (\frac{e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{a b g^{2} + b^{2} g^{2} x} + \frac{\left (B^{2} c + B^{2} d x\right ) \log{\left (\frac{e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}^{2}}{a^{2} d g^{2} - a b c g^{2} + a b d g^{2} x - b^{2} c g^{2} x} - \frac{A^{2} + 4 A B + 8 B^{2}}{a b g^{2} + b^{2} g^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.47634, size = 510, normalized size = 3.92 \begin{align*} -{\left (\frac{B^{2} d}{b^{2} c g^{2} - a b d g^{2}} + \frac{B^{2}}{{\left (b g x + a g\right )} b g}\right )} \log \left (\frac{b^{2}}{\frac{b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac{2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{2 \, b c d g}{b g x + a g} - \frac{2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )^{2} + \frac{4 \,{\left (A B d + 3 \, B^{2} d\right )} \log \left (\frac{b c g}{b g x + a g} - \frac{a d g}{b g x + a g} + d\right )}{b^{2} c g^{2} - a b d g^{2}} - \frac{2 \,{\left (A B + 3 \, B^{2}\right )} \log \left (\frac{b^{2}}{\frac{b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac{2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{2 \, b c d g}{b g x + a g} - \frac{2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )}{{\left (b g x + a g\right )} b g} - \frac{A^{2} + 6 \, A B + 13 \, B^{2}}{{\left (b g x + a g\right )} b g} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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