Optimal. Leaf size=96 \[ \frac{(a+b x) e^{-\frac{A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-1/n} \text{Ei}\left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B g^2 n (c+d x) (b c-a d)} \]
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Rubi [F] time = 0.100225, 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 = {} \[ \int \frac{1}{(c g+d g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{(c g+d g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )} \, dx &=\int \frac{1}{(c g+d g x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )} \, dx\\ \end{align*}
Mathematica [A] time = 0.119399, size = 96, normalized size = 1. \[ \frac{(a+b x) e^{-\frac{A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-1/n} \text{Ei}\left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B g^2 n (c+d x) (b c-a d)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.438, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dgx+cg \right ) ^{2}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \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 (d g x + c g\right )}^{2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.904063, size = 149, normalized size = 1.55 \begin{align*} \frac{e^{\left (-\frac{B \log \left (e\right ) + A}{B n}\right )} \logintegral \left (\frac{{\left (b x + a\right )} e^{\left (\frac{B \log \left (e\right ) + A}{B n}\right )}}{d x + c}\right )}{{\left (B b c - B a d\right )} g^{2} n} \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 (d g x + c g\right )}^{2}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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