Optimal. Leaf size=15 \[ -\frac{1}{c^2 e (d+e x)} \]
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Rubi [A] time = 0.0048716, antiderivative size = 15, 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} \[ -\frac{1}{c^2 e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 32
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\left (c d^2+2 c d e x+c e^2 x^2\right )^2} \, dx &=\int \frac{1}{c^2 (d+e x)^2} \, dx\\ &=\frac{\int \frac{1}{(d+e x)^2} \, dx}{c^2}\\ &=-\frac{1}{c^2 e (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.0021272, size = 15, normalized size = 1. \[ -\frac{1}{c^2 e (d+e x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 16, normalized size = 1.1 \begin{align*} -{\frac{1}{{c}^{2}e \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08718, size = 26, normalized size = 1.73 \begin{align*} -\frac{1}{c^{2} e^{2} x + c^{2} d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.92588, size = 35, normalized size = 2.33 \begin{align*} -\frac{1}{c^{2} e^{2} x + c^{2} d e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.52727, size = 17, normalized size = 1.13 \begin{align*} - \frac{1}{c^{2} d e + c^{2} e^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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