Optimal. Leaf size=153 \[ -\frac{(c+d x) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )^2}{g^2 (a+b x) (b c-a d)}+\frac{2 A B (c+d x)}{g^2 (a+b x) (b c-a d)}+\frac{2 B^2 (c+d x) \log \left (\frac{e (c+d x)}{a+b x}\right )}{g^2 (a+b x) (b c-a d)}-\frac{2 B^2 (c+d x)}{g^2 (a+b x) (b c-a d)} \]
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Rubi [C] time = 0.762186, antiderivative size = 470, normalized size of antiderivative = 3.07, number of steps used = 26, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{2 B^2 d \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac{2 B^2 d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}+\frac{2 B d \log (a+b x) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{b g^2 (b c-a d)}-\frac{2 B d \log (c+d x) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{b g^2 (b c-a d)}+\frac{2 B \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )}{b g^2 (a+b x)}-\frac{\left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )^2}{b g^2 (a+b x)}+\frac{B^2 d \log ^2(a+b x)}{b g^2 (b c-a d)}+\frac{B^2 d \log ^2(c+d x)}{b g^2 (b c-a d)}-\frac{2 B^2 d \log (a+b x)}{b g^2 (b c-a d)}-\frac{2 B^2 d \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}+\frac{2 B^2 d \log (c+d x)}{b g^2 (b c-a d)}-\frac{2 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac{2 B^2}{b g^2 (a+b x)} \]
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 (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^2} \, dx &=-\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{b g^2 (a+b x)}+\frac{(2 B) \int \frac{(b c-a d) \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{g (a+b x)^2 (c+d x)} \, dx}{b g}\\ &=-\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{b g^2 (a+b x)}+\frac{(2 B (b c-a d)) \int \frac{-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )}{(a+b x)^2 (c+d x)} \, dx}{b g^2}\\ &=-\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{b g^2 (a+b x)}+\frac{(2 B (b c-a d)) \int \left (\frac{b \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{(b c-a d) (a+b x)^2}-\frac{b d \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac{d^2 \left (-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b g^2}\\ &=-\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{b g^2 (a+b x)}+\frac{(2 B) \int \frac{-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )}{(a+b x)^2} \, dx}{g^2}-\frac{(2 B d) \int \frac{-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )}{a+b x} \, dx}{(b c-a d) g^2}+\frac{\left (2 B d^2\right ) \int \frac{-A-B \log \left (\frac{e (c+d x)}{a+b x}\right )}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=\frac{2 B \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b g^2 (a+b x)}+\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{2 B d \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{b g^2 (a+b x)}-\frac{\left (2 B^2\right ) \int \frac{-b c+a d}{(a+b x)^2 (c+d x)} \, dx}{b g^2}-\frac{\left (2 B^2 d\right ) \int \frac{(a+b x) \left (\frac{d e}{a+b x}-\frac{b e (c+d x)}{(a+b x)^2}\right ) \log (a+b x)}{e (c+d x)} \, dx}{b (b c-a d) g^2}+\frac{\left (2 B^2 d\right ) \int \frac{(a+b x) \left (\frac{d e}{a+b x}-\frac{b e (c+d x)}{(a+b x)^2}\right ) \log (c+d x)}{e (c+d x)} \, dx}{b (b c-a d) g^2}\\ &=\frac{2 B \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b g^2 (a+b x)}+\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{2 B d \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{b g^2 (a+b x)}+\frac{\left (2 B^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b g^2}-\frac{\left (2 B^2 d\right ) \int \frac{(a+b x) \left (\frac{d e}{a+b x}-\frac{b e (c+d x)}{(a+b x)^2}\right ) \log (a+b x)}{c+d x} \, dx}{b (b c-a d) e g^2}+\frac{\left (2 B^2 d\right ) \int \frac{(a+b x) \left (\frac{d e}{a+b x}-\frac{b e (c+d x)}{(a+b x)^2}\right ) \log (c+d x)}{c+d x} \, dx}{b (b c-a d) e g^2}\\ &=\frac{2 B \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b g^2 (a+b x)}+\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{2 B d \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{b g^2 (a+b x)}+\frac{\left (2 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 (2 B^2 d\right ) \int \left (-\frac{b e \log (a+b x)}{a+b x}+\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{b (b c-a d) e g^2}+\frac{\left (2 B^2 d\right ) \int \left (-\frac{b e \log (c+d x)}{a+b x}+\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{b (b c-a d) e g^2}\\ &=-\frac{2 B^2}{b g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{b (b c-a d) g^2}+\frac{2 B^2 d \log (c+d x)}{b (b c-a d) g^2}+\frac{2 B \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b g^2 (a+b x)}+\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{2 B d \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{b g^2 (a+b x)}+\frac{\left (2 B^2 d\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{(b c-a d) g^2}-\frac{\left (2 B^2 d\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{(b c-a d) g^2}-\frac{\left (2 B^2 d^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b (b c-a d) g^2}+\frac{\left (2 B^2 d^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=-\frac{2 B^2}{b g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{b (b c-a d) g^2}+\frac{2 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac{2 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{2 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{2 B \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b g^2 (a+b x)}+\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{2 B d \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{b g^2 (a+b x)}+\frac{\left (2 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 (2 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 (2 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 (2 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{2 B^2}{b g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{b (b c-a d) g^2}+\frac{B^2 d \log ^2(a+b x)}{b (b c-a d) g^2}+\frac{2 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac{2 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{B^2 d \log ^2(c+d x)}{b (b c-a d) g^2}-\frac{2 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{2 B \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b g^2 (a+b x)}+\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{2 B d \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{b g^2 (a+b x)}+\frac{\left (2 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 (2 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{2 B^2}{b g^2 (a+b x)}-\frac{2 B^2 d \log (a+b x)}{b (b c-a d) g^2}+\frac{B^2 d \log ^2(a+b x)}{b (b c-a d) g^2}+\frac{2 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac{2 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{B^2 d \log ^2(c+d x)}{b (b c-a d) g^2}-\frac{2 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{2 B \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b g^2 (a+b x)}+\frac{2 B d \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{2 B d \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )^2}{b g^2 (a+b x)}-\frac{2 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{2 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 = 0.494578, size = 314, normalized size = 2.05 \[ -\frac{\frac{B \left (-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+B d (a+b x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 (b c-a d) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )-2 d (a+b x) \log (a+b x) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )+2 d (a+b x) \log (c+d x) \left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )+2 B (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )}{b c-a d}+\left (B \log \left (\frac{e (c+d x)}{a+b x}\right )+A\right )^2}{b g^2 (a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 1251, normalized size = 8.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.29471, size = 562, normalized size = 3.67 \begin{align*}{\left (2 \,{\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{d e x}{b x + a} + \frac{c e}{b x + a}\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{d e x}{b x + a} + \frac{c e}{b x + a}\right )}{b^{2} g^{2} x + a b g^{2}} - \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 )} - \frac{B^{2} \log \left (\frac{d e x}{b x + a} + \frac{c e}{b x + a}\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 = 1.01157, size = 319, normalized size = 2.08 \begin{align*} -\frac{{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b c -{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a d +{\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac{d e x + c e}{b x + a}\right )^{2} + 2 \,{\left ({\left (A B - B^{2}\right )} b d x +{\left (A B - B^{2}\right )} b c\right )} \log \left (\frac{d e x + c e}{b x + a}\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 = 3.7579, size = 430, normalized size = 2.81 \begin{align*} \frac{2 B d \left (A - B\right ) \log{\left (x + \frac{2 A B a d^{2} + 2 A B b c d - 2 B^{2} a d^{2} - 2 B^{2} b c d - \frac{2 B a^{2} d^{3} \left (A - B\right )}{a d - b c} + \frac{4 B a b c d^{2} \left (A - B\right )}{a d - b c} - \frac{2 B b^{2} c^{2} d \left (A - B\right )}{a d - b c}}{4 A B b d^{2} - 4 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} - \frac{2 B d \left (A - B\right ) \log{\left (x + \frac{2 A B a d^{2} + 2 A B b c d - 2 B^{2} a d^{2} - 2 B^{2} b c d + \frac{2 B a^{2} d^{3} \left (A - B\right )}{a d - b c} - \frac{4 B a b c d^{2} \left (A - B\right )}{a d - b c} + \frac{2 B b^{2} c^{2} d \left (A - B\right )}{a d - b c}}{4 A B b d^{2} - 4 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac{\left (- 2 A B + 2 B^{2}\right ) \log{\left (\frac{e \left (c + d x\right )}{a + b x} \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 (c + d x\right )}{a + b x} \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} - 2 A B + 2 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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (d x + c\right )} e}{b x + a}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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