Optimal. Leaf size=91 \[ \frac{e^{\frac{A}{2 B}} (c+d x) \sqrt{\frac{e (a+b x)^2}{(c+d x)^2}} \text{Ei}\left (-\frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{2 B}\right )}{2 B g^2 (a+b x) (b c-a d)} \]
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Rubi [F] time = 0.0867152, 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 = {} \[ \int \frac{1}{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx \]
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)^2}{(c+d x)^2}\right )\right )} \, dx &=\int \frac{1}{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx\\ \end{align*}
Mathematica [F] time = 0.0738722, size = 0, normalized size = 0. \[ \int \frac{1}{(a g+b g x)^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 1.091, 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 ) ^{2}}{ \left ( dx+c \right ) ^{2}}} \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 )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}}\,{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{1}{A b^{2} g^{2} x^{2} + 2 \, A a b g^{2} x + A a^{2} g^{2} +{\left (B b^{2} g^{2} x^{2} + 2 \, B a b g^{2} x + B a^{2} g^{2}\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}, 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{1}{{\left (b g x + a g\right )}^{2}{\left (B \log \left (\frac{{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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