Optimal. Leaf size=36 \[ -\frac{\log \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]
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Rubi [A] time = 0.316718, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 53, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038, Rules used = {12, 6684} \[ -\frac{\log \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )}{n (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 6684
Rubi steps
\begin{align*} \int \frac{e \left (\frac{a+b x}{c+d x}\right )^n}{(a+b x) (c+d x) \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx &=e \int \frac{\left (\frac{a+b x}{c+d x}\right )^n}{(a+b x) (c+d x) \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )} \, dx\\ &=-\frac{\log \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(b c-a d) n}\\ \end{align*}
Mathematica [A] time = 0.0914007, size = 38, normalized size = 1.06 \[ -\frac{e \log \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b c e n-a d e n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.121, size = 37, normalized size = 1. \begin{align*}{\frac{1}{n \left ( ad-bc \right ) }\ln \left ( e{{\rm e}^{n\ln \left ({\frac{bx+a}{dx+c}} \right ) }}-1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01397, size = 78, normalized size = 2.17 \begin{align*} -e{\left (\frac{\log \left (-{\left (b x + a\right )}^{n} e +{\left (d x + c\right )}^{n}\right )}{b c e n - a d e n} - \frac{\log \left (d x + c\right )}{b c e - a d e}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.34046, size = 72, normalized size = 2. \begin{align*} -\frac{\log \left (e \left (\frac{b x + a}{d x + c}\right )^{n} - 1\right )}{{\left (b c - a d\right )} 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{e \left (\frac{b x + a}{d x + c}\right )^{n}}{{\left (b x + a\right )}{\left (d x + c\right )}{\left (e \left (\frac{b x + a}{d x + c}\right )^{n} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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