Optimal. Leaf size=129 \[ \frac{8 B^2 (b c-a d) \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{b d}+\frac{4 B (b c-a d) \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b d}+\frac{(a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{b} \]
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Rubi [A] time = 0.773903, antiderivative size = 252, normalized size of antiderivative = 1.95, number of steps used = 22, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {2523, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac{8 a B^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b}+\frac{8 B^2 c \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d}+\frac{4 a B \log (a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b}-\frac{4 B c \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{d}+x \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2+\frac{8 B^2 c \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d}+\frac{8 a B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b}-\frac{4 a B^2 \log ^2(a+b x)}{b}-\frac{4 B^2 c \log ^2(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 2523
Rule 12
Rule 2528
Rule 2524
Rule 2418
Rule 2390
Rule 2301
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2 \, dx &=x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2-(2 B) \int \frac{2 (b c-a d) x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(a+b x) (c+d x)} \, dx\\ &=x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2-(4 B (b c-a d)) \int \frac{x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(a+b x) (c+d x)} \, dx\\ &=x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2-(4 B (b c-a d)) \int \left (-\frac{a \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) (a+b x)}+\frac{c \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) (c+d x)}\right ) \, dx\\ &=x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2+(4 a B) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx-(4 B c) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx\\ &=\frac{4 a B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b}+x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2-\frac{4 B c \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d}-\frac{\left (4 a B^2\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}+\frac{\left (4 B^2 c\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}{d}\\ &=\frac{4 a B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b}+x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2-\frac{4 B c \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d}-\frac{\left (4 a B^2\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 e}+\frac{\left (4 B^2 c\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}{d e}\\ &=\frac{4 a B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b}+x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2-\frac{4 B c \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d}-\frac{\left (4 a B^2\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 e}+\frac{\left (4 B^2 c\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}{d e}\\ &=\frac{4 a B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b}+x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2-\frac{4 B c \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d}-\left (8 a B^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx-\left (8 B^2 c\right ) \int \frac{\log (c+d x)}{c+d x} \, dx+\frac{\left (8 b B^2 c\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{d}+\frac{\left (8 a B^2 d\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b}\\ &=\frac{4 a B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b}+x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac{8 B^2 c \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac{4 B c \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d}+\frac{8 a B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b}-\left (8 a B^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx-\frac{\left (8 a B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b}-\left (8 B^2 c\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx-\frac{\left (8 B^2 c\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{d}\\ &=-\frac{4 a B^2 \log ^2(a+b x)}{b}+\frac{4 a B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b}+x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac{8 B^2 c \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac{4 B c \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d}-\frac{4 B^2 c \log ^2(c+d x)}{d}+\frac{8 a B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b}-\frac{\left (8 a B^2\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}-\frac{\left (8 B^2 c\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{d}\\ &=-\frac{4 a B^2 \log ^2(a+b x)}{b}+\frac{4 a B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b}+x \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac{8 B^2 c \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{d}-\frac{4 B c \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{d}-\frac{4 B^2 c \log ^2(c+d x)}{d}+\frac{8 a B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b}+\frac{8 a B^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b}+\frac{8 B^2 c \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.162735, size = 220, normalized size = 1.71 \[ \frac{4 B \left (-a B d \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+b B c \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+a d \log (a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )-b c \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )\right )}{b d}+x \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2 \]
Antiderivative was successfully verified.
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Maple [F] time = 1.319, size = 0, normalized size = 0. \begin{align*} \int \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} 2 \,{\left (x \log \left (\frac{{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + \frac{2 \,{\left (\frac{a e \log \left (b x + a\right )}{b} - \frac{c e \log \left (d x + c\right )}{d}\right )}}{e}\right )} A B + A^{2} x + B^{2}{\left (\frac{4 \,{\left (b d x \log \left (b x + a\right )^{2} +{\left (b d x + b c\right )} \log \left (d x + c\right )^{2} -{\left (b d x \log \left (e\right ) + 2 \,{\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )}}{b d} + \int \frac{{\left (\log \left (e\right )^{2} + 4 \, \log \left (e\right )\right )} b^{2} d x^{2} + a b c \log \left (e\right )^{2} +{\left (b^{2} c \log \left (e\right )^{2} +{\left (\log \left (e\right )^{2} + 4 \, \log \left (e\right )\right )} a b d\right )} x + 4 \,{\left (b^{2} d x^{2} \log \left (e\right ) + a b c \log \left (e\right ) + 2 \, a^{2} d +{\left (a b d{\left (\log \left (e\right ) + 4\right )} + b^{2} c{\left (\log \left (e\right ) - 2\right )}\right )} x\right )} \log \left (b x + a\right )}{b^{2} d x^{2} + a b c +{\left (b^{2} c + a b d\right )} x}\,{d x}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (B^{2} \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 \, A B \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 ) + A^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\left (B \log \left (\frac{{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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