Optimal. Leaf size=50 \[ \frac{e e^{A/B} \text{Ei}\left (-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{B}\right )}{B g^2 (b c-a d)} \]
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Rubi [F] time = 0.0900594, 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 = {} \[ \int \frac{1}{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )} \, dx \]
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 )} \, 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 )} \, dx\\ \end{align*}
Mathematica [A] time = 0.102231, size = 52, normalized size = 1.04 \[ \frac{e e^{A/B} \text{Ei}\left (-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{B}\right )}{b B c g^2-a B d g^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.187, 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 ) ^{-1}}\, 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*} \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 )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.03048, size = 108, normalized size = 2.16 \begin{align*} \frac{e e^{\frac{A}{B}} \logintegral \left (\frac{{\left (d x + c\right )} e^{\left (-\frac{A}{B}\right )}}{b e x + a e}\right )}{{\left (B b c - B a d\right )} g^{2}} \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 )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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