Optimal. Leaf size=32 \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{e x}{d}+1\right )}{c d m} \]
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Rubi [A] time = 0.0151232, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {626, 12, 65} \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{e x}{d}+1\right )}{c d m} \]
Antiderivative was successfully verified.
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Rule 626
Rule 12
Rule 65
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{c d x+c e x^2} \, dx &=\int \frac{(d+e x)^{-1+m}}{c x} \, dx\\ &=\frac{\int \frac{(d+e x)^{-1+m}}{x} \, dx}{c}\\ &=-\frac{(d+e x)^m \, _2F_1\left (1,m;1+m;1+\frac{e x}{d}\right )}{c d m}\\ \end{align*}
Mathematica [A] time = 0.0084562, size = 32, normalized size = 1. \[ -\frac{(d+e x)^m \, _2F_1\left (1,m;m+1;\frac{e x}{d}+1\right )}{c d m} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.547, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{m}}{ce{x}^{2}+cdx}}\, 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 e x^{2} + c d 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 e x^{2} + c d 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*} \frac{\int \frac{\left (d + e x\right )^{m}}{d x + e x^{2}}\, dx}{c} \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 e x^{2} + c d x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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