Optimal. Leaf size=54 \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )} \]
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Rubi [A] time = 0.0243076, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057, Rules used = {626, 68} \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Rule 626
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac{(d+e x)^{-1+m}}{a e+c d x} \, dx\\ &=-\frac{(d+e x)^m \, _2F_1\left (1,m;1+m;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) m}\\ \end{align*}
Mathematica [A] time = 0.012819, size = 54, normalized size = 1. \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{m \left (c d^2-a e^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.179, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}\, 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 (e x + d\right )}^{m}}{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e x + d\right )}^{m}}{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (d + e x\right )^{m}}{\left (d + e x\right ) \left (a e + c d x\right )}\, dx \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 x + d\right )}^{m}}{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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