Optimal. Leaf size=103 \[ -\frac{e e^{A/B} \text{Ei}\left (-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{B}\right )}{B^2 g^2 (b c-a d)}-\frac{c+d x}{B g^2 (a+b x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )} \]
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Rubi [F] time = 0.094921, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2} \, dx &=\int \frac{1}{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2} \, dx\\ \end{align*}
Mathematica [A] time = 0.187646, size = 87, normalized size = 0.84 \[ \frac{e e^{A/B} \text{Ei}\left (-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{B}\right )+\frac{B (c+d x)}{(a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}}{B^2 g^2 (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.274, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{2}} \left ( A+B\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{d x + c}{{\left (a b c g^{2} - a^{2} d g^{2}\right )} A B +{\left (a b c g^{2} \log \left (e\right ) - a^{2} d g^{2} \log \left (e\right )\right )} B^{2} +{\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} A B +{\left (b^{2} c g^{2} \log \left (e\right ) - a b d g^{2} \log \left (e\right )\right )} B^{2}\right )} x +{\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} B^{2} x +{\left (a b c g^{2} - a^{2} d g^{2}\right )} B^{2}\right )} \log \left (b x + a\right ) -{\left ({\left (b^{2} c g^{2} - a b d g^{2}\right )} B^{2} x +{\left (a b c g^{2} - a^{2} d g^{2}\right )} B^{2}\right )} \log \left (d x + c\right )} + \int -\frac{1}{B^{2} a^{2} g^{2} \log \left (e\right ) + A B a^{2} g^{2} +{\left (B^{2} b^{2} g^{2} \log \left (e\right ) + A B b^{2} g^{2}\right )} x^{2} + 2 \,{\left (B^{2} a b g^{2} \log \left (e\right ) + A B a b g^{2}\right )} x +{\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (b x + a\right ) -{\left (B^{2} b^{2} g^{2} x^{2} + 2 \, B^{2} a b g^{2} x + B^{2} a^{2} g^{2}\right )} \log \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.02172, size = 423, normalized size = 4.11 \begin{align*} -\frac{B d x + B c +{\left ({\left (B b e x + B a e\right )} e^{\frac{A}{B}} \log \left (\frac{b e x + a e}{d x + c}\right ) +{\left (A b e x + A a e\right )} e^{\frac{A}{B}}\right )} \logintegral \left (\frac{{\left (d x + c\right )} e^{\left (-\frac{A}{B}\right )}}{b e x + a e}\right )}{{\left (A B^{2} b^{2} c - A B^{2} a b d\right )} g^{2} x +{\left (A B^{2} a b c - A B^{2} a^{2} d\right )} g^{2} +{\left ({\left (B^{3} b^{2} c - B^{3} a b d\right )} g^{2} x +{\left (B^{3} a b c - B^{3} a^{2} d\right )} g^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\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{1}{{\left (b g x + a g\right )}^{2}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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