Optimal. Leaf size=125 \[ \frac{(a+b x) e^{-\frac{A (m+1)}{B n}} (g (a+b x))^m (i (c+d x))^{-m} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-\frac{m+1}{n}} \text{Ei}\left (\frac{(m+1) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B i^2 n (c+d x) (b c-a d)} \]
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Rubi [F] time = 0.736634, 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 = {} \[ \int \frac{(a g+b g x)^m (c i+d i x)^{-2-m}}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{(215 c+215 d x)^{-2-m} (a g+b g x)^m}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx &=\int \frac{(215 c+215 d x)^{-2-m} (a g+b g x)^m}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx\\ \end{align*}
Mathematica [F] time = 0.231827, size = 0, normalized size = 0. \[ \int \frac{(a g+b g x)^m (c i+d i x)^{-2-m}}{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 2.954, size = 0, normalized size = 0. \begin{align*} \int{ \left ( bgx+ag \right ) ^{m} \left ( dix+ci \right ) ^{-2-m} \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{{\left (b g x + a g\right )}^{m}{\left (d i x + c i\right )}^{-m - 2}}{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.496147, size = 227, normalized size = 1.82 \begin{align*} \frac{{\rm Ei}\left (\frac{{\left (B m + B\right )} n \log \left (\frac{b x + a}{d x + c}\right ) + A m +{\left (B m + B\right )} \log \left (e\right ) + A}{B n}\right ) e^{\left (-\frac{{\left (B m + 2 \, B\right )} n \log \left (\frac{i}{g}\right ) + A m +{\left (B m + B\right )} \log \left (e\right ) + A}{B n}\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{{\left (b g x + a g\right )}^{m}{\left (d i x + c i\right )}^{-m - 2}}{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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