Optimal. Leaf size=24 \[ -\frac{(d+e x)^{m-3}}{c^2 e (3-m)} \]
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Rubi [A] time = 0.0101768, 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} \[ -\frac{(d+e x)^{m-3}}{c^2 e (3-m)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{(d+e x)^m}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac{(d+e x)^{-4+m}}{c^2} \, dx\\ &=\frac{\int (d+e x)^{-4+m} \, dx}{c^2}\\ &=-\frac{(d+e x)^{-3+m}}{c^2 e (3-m)}\\ \end{align*}
Mathematica [A] time = 0.0148665, size = 21, normalized size = 0.88 \[ \frac{(d+e x)^{m-3}}{c^2 e (m-3)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 40, normalized size = 1.7 \begin{align*}{\frac{ \left ( ex+d \right ) ^{-1+m}}{ \left ({e}^{2}{x}^{2}+2\,dex+{d}^{2} \right ){c}^{2}e \left ( -3+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.37361, size = 88, normalized size = 3.67 \begin{align*} \frac{{\left (e x + d\right )}^{m}}{c^{2} e^{4}{\left (m - 3\right )} x^{3} + 3 \, c^{2} d e^{3}{\left (m - 3\right )} x^{2} + 3 \, c^{2} d^{2} e^{2}{\left (m - 3\right )} x + c^{2} d^{3} e{\left (m - 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.47713, size = 196, normalized size = 8.17 \begin{align*} \frac{{\left (e x + d\right )}^{m}}{c^{2} d^{3} e m - 3 \, c^{2} d^{3} e +{\left (c^{2} e^{4} m - 3 \, c^{2} e^{4}\right )} x^{3} + 3 \,{\left (c^{2} d e^{3} m - 3 \, c^{2} d e^{3}\right )} x^{2} + 3 \,{\left (c^{2} d^{2} e^{2} m - 3 \, c^{2} d^{2} e^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.14732, size = 136, normalized size = 5.67 \begin{align*} \begin{cases} \frac{x}{c^{2} d} & \text{for}\: e = 0 \wedge m = 3 \\\frac{d^{m} x}{c^{2} d^{4}} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{c^{2} e} & \text{for}\: m = 3 \\\frac{\left (d + e x\right )^{m}}{c^{2} d^{3} e m - 3 c^{2} d^{3} e + 3 c^{2} d^{2} e^{2} m x - 9 c^{2} d^{2} e^{2} x + 3 c^{2} d e^{3} m x^{2} - 9 c^{2} d e^{3} x^{2} + c^{2} e^{4} m x^{3} - 3 c^{2} e^{4} x^{3}} & \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}}{{\left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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