Optimal. Leaf size=52 \[ \frac{e \left (\frac{a+b x}{c+d x}\right )^n}{n (b c-a d) \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )^2} \]
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Rubi [A] time = 2.02456, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 76, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {6741, 12, 6692, 34} \[ \frac{e \left (\frac{a+b x}{c+d x}\right )^n}{n (b c-a d) \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )^2} \]
Antiderivative was successfully verified.
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Rule 6741
Rule 12
Rule 6692
Rule 34
Rubi steps
\begin{align*} \int \frac{e \left (\frac{a+b x}{c+d x}\right )^n+e^2 \left (\frac{a+b x}{c+d x}\right )^{2 n}}{(a+b x) (c+d x) \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )^3} \, dx &=\int \frac{e \left (\frac{a+b x}{c+d x}\right )^n \left (1+e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x) \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )^3} \, dx\\ &=e \int \frac{\left (\frac{a+b x}{c+d x}\right )^n \left (1+e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x) \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )^3} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{1+x}{(1-x)^3} \, dx,x,e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(b c-a d) n}\\ &=\frac{e \left (\frac{a+b x}{c+d x}\right )^n}{(b c-a d) n \left (1-e \left (\frac{a+b x}{c+d x}\right )^n\right )^2}\\ \end{align*}
Mathematica [A] time = 0.265491, size = 52, normalized size = 1. \[ -\frac{e \left (\frac{a+b x}{c+d x}\right )^n}{n (a d-b c) \left (e \left (\frac{a+b x}{c+d x}\right )^n-1\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.276, size = 57, normalized size = 1.1 \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 ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.17927, size = 285, normalized size = 5.48 \begin{align*} \frac{1}{2} \,{\left (\frac{{\left (b x + a\right )}^{2 \, n} e}{{\left (b c e^{2} n - a d e^{2} n\right )}{\left (b x + a\right )}^{2 \, n} +{\left (b c n - a d n\right )}{\left (d x + c\right )}^{2 \, n} - 2 \,{\left (b c e n - a d e n\right )} e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right )\right )}} - \frac{{\left (b x + a\right )}^{2 \, n} e - 2 \, e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right )\right )}}{{\left (b c e^{2} n - a d e^{2} n\right )}{\left (b x + a\right )}^{2 \, n} +{\left (b c n - a d n\right )}{\left (d x + c\right )}^{2 \, n} - 2 \,{\left (b c e n - a d e n\right )} e^{\left (n \log \left (b x + a\right ) + n \log \left (d x + c\right )\right )}}\right )} e \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.27585, size = 182, normalized size = 3.5 \begin{align*} \frac{e \left (\frac{b x + a}{d x + c}\right )^{n}}{{\left (b c - a d\right )} e^{2} n \left (\frac{b x + a}{d x + c}\right )^{2 \, n} - 2 \,{\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{{\left (e \left (\frac{b x + a}{d x + c}\right )^{n} + 1\right )} 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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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