Optimal. Leaf size=24 \[ -\frac{(d+e x)^{m-1}}{c e (1-m)} \]
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Rubi [A] time = 0.0115782, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {27, 12, 32} \[ -\frac{(d+e x)^{m-1}}{c e (1-m)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{c d^2+2 c d e x+c e^2 x^2} \, dx &=\int \frac{(d+e x)^{-2+m}}{c} \, dx\\ &=\frac{\int (d+e x)^{-2+m} \, dx}{c}\\ &=-\frac{(d+e x)^{-1+m}}{c e (1-m)}\\ \end{align*}
Mathematica [A] time = 0.0140982, size = 21, normalized size = 0.88 \[ \frac{(d+e x)^{m-1}}{c e (m-1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 22, normalized size = 0.9 \begin{align*}{\frac{ \left ( ex+d \right ) ^{-1+m}}{ce \left ( -1+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.18196, size = 36, normalized size = 1.5 \begin{align*} \frac{{\left (e x + d\right )}^{m}}{c e^{2}{\left (m - 1\right )} x + c d e{\left (m - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.33927, size = 72, normalized size = 3. \begin{align*} \frac{{\left (e x + d\right )}^{m}}{c d e m - c d e +{\left (c e^{2} m - c e^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.18284, size = 63, normalized size = 2.62 \begin{align*} \begin{cases} \text{NaN} & \text{for}\: d = 0 \wedge e = 0 \wedge m = 1 \\0^{m} \tilde{\infty } x & \text{for}\: d = - e x \\\frac{d^{m} x}{c d^{2}} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{c e} & \text{for}\: m = 1 \\\frac{\left (d + e x\right )^{m}}{c d e m - c d e + c e^{2} m x - c e^{2} x} & \text{otherwise} \end{cases} \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^{2} x^{2} + 2 \, c d e x + c d^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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