Optimal. Leaf size=36 \[ \frac{1}{n (b c-a d) \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )} \]
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Rubi [A] time = 0.366682, 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, 6686} \[ \frac{1}{n (b c-a d) \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )} \]
Antiderivative was successfully verified.
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Rule 12
Rule 6686
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 )^2} \, 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 )^2} \, dx\\ &=\frac{1}{(b c-a d) n \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )}\\ \end{align*}
Mathematica [A] time = 0.103027, size = 35, normalized size = 0.97 \[ \frac{1}{n (a d-b c) \left (e \left (\frac{a+b x}{c+d x}\right )^n-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.171, size = 56, normalized size = 1.6 \begin{align*}{\frac{e}{n \left ( ad-bc \right ) }{{\rm e}^{n\ln \left ({\frac{bx+a}{dx+c}} \right ) }} \left ( e{{\rm e}^{n\ln \left ({\frac{bx+a}{dx+c}} \right ) }}-1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04494, size = 70, normalized size = 1.94 \begin{align*} -\frac{{\left (b x + a\right )}^{n} e}{{\left (b c e n - a d e n\right )}{\left (b x + a\right )}^{n} -{\left (b c n - a d n\right )}{\left (d x + c\right )}^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.47833, size = 84, normalized size = 2.33 \begin{align*} -\frac{1}{{\left (b c - a d\right )} e n \left (\frac{b x + a}{d x + c}\right )^{n} -{\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 )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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