Optimal. Leaf size=61 \[ -\frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(1-m) \left (c d^2-a e^2\right )^2} \]
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Rubi [A] time = 0.0223567, antiderivative size = 61, 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} \[ -\frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{(1-m) \left (c d^2-a e^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 626
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 )^2} \, dx &=\int \frac{(d+e x)^{-2+m}}{(a e+c d x)^2} \, dx\\ &=-\frac{e (d+e x)^{-1+m} \, _2F_1\left (2,-1+m;m;\frac{c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right )^2 (1-m)}\\ \end{align*}
Mathematica [A] time = 0.0230798, size = 59, normalized size = 0.97 \[ \frac{e (d+e x)^{m-1} \, _2F_1\left (2,m-1;m;-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{(m-1) \left (a e^2-c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 1.269, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ \left ( ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2} \right ) ^{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}}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{2}}\,{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^{2} d^{2} e^{2} x^{4} + a^{2} d^{2} e^{2} + 2 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{3} +{\left (c^{2} d^{4} + 4 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x}, 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 (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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