Optimal. Leaf size=107 \[ \frac{b e^2 e^{\frac{2 A}{B}} \text{Ei}\left (-\frac{2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{B}\right )}{B g^3 (b c-a d)^2}-\frac{d e e^{A/B} \text{Ei}\left (-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{B}\right )}{B g^3 (b c-a d)^2} \]
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Rubi [F] time = 0.0729636, 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 = {} \[ \int \frac{1}{(a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{(a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )} \, dx &=\int \frac{1}{(a g+b g x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )} \, dx\\ \end{align*}
Mathematica [A] time = 0.165399, size = 89, normalized size = 0.83 \[ \frac{e e^{A/B} \left (b e e^{A/B} \text{Ei}\left (-\frac{2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{B}\right )-d \text{Ei}\left (-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{B}\right )\right )}{B g^3 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.283, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{3}} \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 )}^{3}{\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 = 1.00978, size = 288, normalized size = 2.69 \begin{align*} \frac{b e^{2} e^{\left (\frac{2 \, A}{B}\right )} \logintegral \left (\frac{{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} e^{\left (-\frac{2 \, A}{B}\right )}}{b^{2} e^{2} x^{2} + 2 \, a b e^{2} x + a^{2} e^{2}}\right ) - d 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^{2} c^{2} - 2 \, B a b c d + B a^{2} d^{2}\right )} g^{3}} \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 )}^{3}{\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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