Optimal. Leaf size=109 \[ \frac{d e^{-\frac{A}{B}} \text{Ei}\left (\frac{A+B \log \left (\frac{e (c+d x)}{a+b x}\right )}{B}\right )}{B e g^3 (b c-a d)^2}-\frac{b e^{-\frac{2 A}{B}} \text{Ei}\left (\frac{2 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{B}\right )}{B e^2 g^3 (b c-a d)^2} \]
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Rubi [F] time = 0.0719976, 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 = {} \[ \int \frac{1}{(a g+b g x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )} \, dx \]
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 (c+d x)}{a+b x}\right )\right )} \, dx &=\int \frac{1}{(a g+b g x)^3 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )} \, dx\\ \end{align*}
Mathematica [A] time = 0.157746, size = 89, normalized size = 0.82 \[ \frac{e^{-\frac{2 A}{B}} \left (d e e^{A/B} \text{Ei}\left (\frac{A}{B}+\log \left (\frac{e (c+d x)}{a+b x}\right )\right )-b \text{Ei}\left (\frac{2 \left (A+B \log \left (\frac{e (c+d x)}{a+b x}\right )\right )}{B}\right )\right )}{B e^2 g^3 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.425, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{3}} \left ( A+B\ln \left ({\frac{e \left ( dx+c \right ) }{bx+a}} \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 (d x + c\right )} e}{b x + a}\right ) + A\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.976792, size = 286, normalized size = 2.62 \begin{align*} \frac{{\left (d e e^{\frac{A}{B}} \logintegral \left (\frac{{\left (d e x + c e\right )} e^{\frac{A}{B}}}{b x + a}\right ) - b \logintegral \left (\frac{{\left (d^{2} e^{2} x^{2} + 2 \, c d e^{2} x + c^{2} e^{2}\right )} e^{\left (\frac{2 \, A}{B}\right )}}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )\right )} e^{\left (-\frac{2 \, A}{B}\right )}}{{\left (B b^{2} c^{2} - 2 \, B a b c d + B a^{2} d^{2}\right )} e^{2} 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 (d x + c\right )} e}{b x + a}\right ) + A\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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