Optimal. Leaf size=13 \[ \frac{\log (d+e x)}{c e} \]
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Rubi [A] time = 0.0040553, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {27, 12, 31} \[ \frac{\log (d+e x)}{c e} \]
Antiderivative was successfully verified.
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Rule 27
Rule 12
Rule 31
Rubi steps
\begin{align*} \int \frac{d+e x}{c d^2+2 c d e x+c e^2 x^2} \, dx &=\int \frac{1}{c (d+e x)} \, dx\\ &=\frac{\int \frac{1}{d+e x} \, dx}{c}\\ &=\frac{\log (d+e x)}{c e}\\ \end{align*}
Mathematica [A] time = 0.0017076, size = 16, normalized size = 1.23 \[ \frac{\log (c d+c e x)}{c e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 14, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( ex+d \right ) }{ce}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.17815, size = 39, normalized size = 3. \begin{align*} \frac{\log \left (c e^{2} x^{2} + 2 \, c d e x + c d^{2}\right )}{2 \, c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91153, size = 27, normalized size = 2.08 \begin{align*} \frac{\log \left (e x + d\right )}{c e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.137631, size = 12, normalized size = 0.92 \begin{align*} \frac{\log{\left (c d + c e x \right )}}{c e} \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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