Optimal. Leaf size=99 \[ \frac{(d+e x)^m (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \, _2F_1\left (1,1-m;2-m;-\frac{g (a e+c d x)}{c d f-a e g}\right )}{(1-m) (c d f-a e g)} \]
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Rubi [A] time = 0.0612081, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 44, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {891, 68} \[ \frac{(d+e x)^m (a e+c d x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{-m} \, _2F_1\left (1,1-m;2-m;-\frac{g (a e+c d x)}{c d f-a e g}\right )}{(1-m) (c d f-a e g)} \]
Antiderivative was successfully verified.
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Rule 891
Rule 68
Rubi steps
\begin{align*} \int \frac{(d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}}{f+g x} \, dx &=\left ((a e+c d x)^m (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m}\right ) \int \frac{(a e+c d x)^{-m}}{f+g x} \, dx\\ &=\frac{(a e+c d x) (d+e x)^m \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{-m} \, _2F_1\left (1,1-m;2-m;-\frac{g (a e+c d x)}{c d f-a e g}\right )}{(c d f-a e g) (1-m)}\\ \end{align*}
Mathematica [A] time = 0.0280362, size = 82, normalized size = 0.83 \[ -\frac{(d+e x)^{m-1} ((d+e x) (a e+c d x))^{1-m} \, _2F_1\left (1,1-m;2-m;\frac{g (a e+c d x)}{a e g-c d f}\right )}{(m-1) (c d f-a e g)} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.767, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( gx+f \right ) \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{m}}}\, 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}}{{\left (g x + f\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{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}}{{\left (g x + f\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}, x\right ) \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 x + d\right )}^{m}}{{\left (g x + f\right )}{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{m}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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