Optimal. Leaf size=150 \[ -\frac{\text{PolyLog}\left (2,1-\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{g (b c-a d)}+\frac{2 \text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )}{g (b c-a d)}-\frac{2 \text{PolyLog}\left (4,1-\frac{b c-a d}{b (c+d x)}\right )}{g (b c-a d)} \]
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Rubi [A] time = 0.247639, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 55, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.055, Rules used = {2506, 2508, 6610} \[ -\frac{\text{PolyLog}\left (2,1-\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{g (b c-a d)}+\frac{2 \text{PolyLog}\left (3,1-\frac{b c-a d}{b (c+d x)}\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )}{g (b c-a d)}-\frac{2 \text{PolyLog}\left (4,1-\frac{b c-a d}{b (c+d x)}\right )}{g (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 2506
Rule 2508
Rule 6610
Rubi steps
\begin{align*} \int \frac{\log \left (\frac{b c-a d}{b (c+d x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x) (a g+b g x)} \, dx &=-\frac{\log ^2\left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(b c-a d) g}+\frac{2 \int \frac{\log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{g}\\ &=-\frac{\log ^2\left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(b c-a d) g}+\frac{2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(b c-a d) g}-\frac{2 \int \frac{\text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{g}\\ &=-\frac{\log ^2\left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_2\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(b c-a d) g}+\frac{2 \log \left (\frac{e (a+b x)}{c+d x}\right ) \text{Li}_3\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(b c-a d) g}-\frac{2 \text{Li}_4\left (1-\frac{b c-a d}{b (c+d x)}\right )}{(b c-a d) g}\\ \end{align*}
Mathematica [A] time = 0.0376798, size = 110, normalized size = 0.73 \[ \frac{-\text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right ) \log ^2\left (\frac{e (a+b x)}{c+d x}\right )+2 \text{PolyLog}\left (3,\frac{d (a+b x)}{b (c+d x)}\right ) \log \left (\frac{e (a+b x)}{c+d x}\right )-2 \text{PolyLog}\left (4,\frac{d (a+b x)}{b (c+d x)}\right )}{g (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dx+c \right ) \left ( bgx+ag \right ) }\ln \left ({\frac{-ad+bc}{b \left ( dx+c \right ) }} \right ) \left ( \ln \left ({\frac{e \left ( bx+a \right ) }{dx+c}} \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*} -\frac{4 \, \log \left (b x + a\right ) \log \left (d x + c\right )^{3} - \log \left (d x + c\right )^{4}}{4 \,{\left (b c g - a d g\right )}} - \int \frac{{\left ({\left (d \log \left (b c - a d\right ) - d \log \left (b\right )\right )} a -{\left (c \log \left (b c - a d\right ) - c \log \left (b\right )\right )} b\right )} \log \left (b x + a\right )^{2} +{\left ({\left (d \log \left (b c - a d\right ) - d \log \left (b\right ) + 2 \, d \log \left (e\right )\right )} a -{\left (c{\left (\log \left (b c - a d\right ) + 2 \, \log \left (e\right )\right )} - c \log \left (b\right )\right )} b -{\left (3 \, b d x + 2 \, b c + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )^{2} +{\left (d \log \left (b c - a d\right ) \log \left (e\right )^{2} - d \log \left (b\right ) \log \left (e\right )^{2}\right )} a -{\left (c \log \left (b c - a d\right ) \log \left (e\right )^{2} - c \log \left (b\right ) \log \left (e\right )^{2}\right )} b + 2 \,{\left ({\left (d \log \left (b c - a d\right ) \log \left (e\right ) - d \log \left (b\right ) \log \left (e\right )\right )} a -{\left (c \log \left (b c - a d\right ) \log \left (e\right ) - c \log \left (b\right ) \log \left (e\right )\right )} b\right )} \log \left (b x + a\right ) +{\left ({\left (b c - a d\right )} \log \left (b x + a\right )^{2} -{\left (2 \, d \log \left (b c - a d\right ) \log \left (e\right ) - 2 \, d \log \left (b\right ) \log \left (e\right ) + d \log \left (e\right )^{2}\right )} a -{\left (2 \, c \log \left (b\right ) \log \left (e\right ) -{\left (2 \, \log \left (b c - a d\right ) \log \left (e\right ) + \log \left (e\right )^{2}\right )} c\right )} b - 2 \,{\left ({\left (d \log \left (b c - a d\right ) - d \log \left (b\right ) + d \log \left (e\right )\right )} a -{\left (c{\left (\log \left (b c - a d\right ) + \log \left (e\right )\right )} - c \log \left (b\right )\right )} b\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b c^{2} g - a^{2} c d g +{\left (b^{2} c d g - a b d^{2} g\right )} x^{2} +{\left (b^{2} c^{2} g - a^{2} d^{2} g\right )} x}\,{d x} \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 (\frac{\log \left (\frac{b c - a d}{b d x + b c}\right ) \log \left (\frac{b e x + a e}{d x + c}\right )^{2}}{b d g x^{2} + a c g +{\left (b c + a d\right )} g x}, 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 \frac{\log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right )^{2} \log \left (\frac{b c - a d}{{\left (d x + c\right )} b}\right )}{{\left (b g x + a g\right )}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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