Optimal. Leaf size=24 \[ -\frac{(d+e x)^{m-5}}{c^3 e (5-m)} \]
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Rubi [A] time = 0.0102396, 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} \[ -\frac{(d+e x)^{m-5}}{c^3 e (5-m)} \]
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 )^3} \, dx &=\int \frac{(d+e x)^{-6+m}}{c^3} \, dx\\ &=\frac{\int (d+e x)^{-6+m} \, dx}{c^3}\\ &=-\frac{(d+e x)^{-5+m}}{c^3 e (5-m)}\\ \end{align*}
Mathematica [A] time = 0.0144844, size = 21, normalized size = 0.88 \[ \frac{(d+e x)^{m-5}}{c^3 e (m-5)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, 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 ) ^{2}{c}^{3}e \left ( -5+m \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20799, size = 134, normalized size = 5.58 \begin{align*} \frac{{\left (e x + d\right )}^{m}}{c^{3} e^{6}{\left (m - 5\right )} x^{5} + 5 \, c^{3} d e^{5}{\left (m - 5\right )} x^{4} + 10 \, c^{3} d^{2} e^{4}{\left (m - 5\right )} x^{3} + 10 \, c^{3} d^{3} e^{3}{\left (m - 5\right )} x^{2} + 5 \, c^{3} d^{4} e^{2}{\left (m - 5\right )} x + c^{3} d^{5} e{\left (m - 5\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.49311, size = 306, normalized size = 12.75 \begin{align*} \frac{{\left (e x + d\right )}^{m}}{c^{3} d^{5} e m - 5 \, c^{3} d^{5} e +{\left (c^{3} e^{6} m - 5 \, c^{3} e^{6}\right )} x^{5} + 5 \,{\left (c^{3} d e^{5} m - 5 \, c^{3} d e^{5}\right )} x^{4} + 10 \,{\left (c^{3} d^{2} e^{4} m - 5 \, c^{3} d^{2} e^{4}\right )} x^{3} + 10 \,{\left (c^{3} d^{3} e^{3} m - 5 \, c^{3} d^{3} e^{3}\right )} x^{2} + 5 \,{\left (c^{3} d^{4} e^{2} m - 5 \, c^{3} d^{4} e^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.70306, size = 201, normalized size = 8.38 \begin{align*} \begin{cases} \frac{x}{c^{3} d} & \text{for}\: e = 0 \wedge m = 5 \\\frac{d^{m} x}{c^{3} d^{6}} & \text{for}\: e = 0 \\\frac{\log{\left (\frac{d}{e} + x \right )}}{c^{3} e} & \text{for}\: m = 5 \\\frac{\left (d + e x\right )^{m}}{c^{3} d^{5} e m - 5 c^{3} d^{5} e + 5 c^{3} d^{4} e^{2} m x - 25 c^{3} d^{4} e^{2} x + 10 c^{3} d^{3} e^{3} m x^{2} - 50 c^{3} d^{3} e^{3} x^{2} + 10 c^{3} d^{2} e^{4} m x^{3} - 50 c^{3} d^{2} e^{4} x^{3} + 5 c^{3} d e^{5} m x^{4} - 25 c^{3} d e^{5} x^{4} + c^{3} e^{6} m x^{5} - 5 c^{3} e^{6} x^{5}} & \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 )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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