Optimal. Leaf size=199 \[ \frac{b (a+b x) e^{-\frac{A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-1/n} \text{Ei}\left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )}{B g^3 n (c+d x) (b c-a d)^2}-\frac{d (a+b x)^2 e^{-\frac{2 A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-2/n} \text{Ei}\left (\frac{2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )}{B g^3 n (c+d x)^2 (b c-a d)^2} \]
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Rubi [F] time = 0.0828902, 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 = {} \[ \int \frac{1}{(c g+d g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{(c g+d g x)^3 \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)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )} \, dx\\ \end{align*}
Mathematica [A] time = 0.290127, size = 174, normalized size = 0.87 \[ \frac{(a+b x) e^{-\frac{2 A}{B n}} \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{-2/n} \left (b e^{\frac{A}{B n}} (c+d x) \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )^{\frac{1}{n}} \text{Ei}\left (\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{B n}\right )-d (a+b x) \text{Ei}\left (\frac{2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{B n}\right )\right )}{B g^3 n (c+d x)^2 (b c-a d)^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.435, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dgx+cg \right ) ^{3}} \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 )}^{3}{\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 = 0.88901, size = 351, normalized size = 1.76 \begin{align*} \frac{{\left (b 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 ) - d \logintegral \left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e^{\left (\frac{2 \,{\left (B \log \left (e\right ) + A\right )}}{B n}\right )}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )\right )} e^{\left (-\frac{2 \,{\left (B \log \left (e\right ) + A\right )}}{B n}\right )}}{{\left (B b^{2} c^{2} - 2 \, B a b c d + B a^{2} d^{2}\right )} g^{3} 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 )}^{3}{\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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