Optimal. Leaf size=16 \[ \frac{\log (a e+c d x)}{c d} \]
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Rubi [A] time = 0.0061629, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061, Rules used = {626, 31} \[ \frac{\log (a e+c d x)}{c d} \]
Antiderivative was successfully verified.
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Rule 626
Rule 31
Rubi steps
\begin{align*} \int \frac{d+e x}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac{1}{a e+c d x} \, dx\\ &=\frac{\log (a e+c d x)}{c d}\\ \end{align*}
Mathematica [A] time = 0.0015131, size = 16, normalized size = 1. \[ \frac{\log (a e+c d x)}{c d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 17, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( cdx+ae \right ) }{cd}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0236, size = 22, normalized size = 1.38 \begin{align*} \frac{\log \left (c d x + a e\right )}{c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.62955, size = 32, normalized size = 2. \begin{align*} \frac{\log \left (c d x + a e\right )}{c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.083825, size = 12, normalized size = 0.75 \begin{align*} \frac{\log{\left (a e + c d x \right )}}{c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22298, size = 170, normalized size = 10.62 \begin{align*} \frac{{\left (c d^{2} - a e^{2}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c d} + \frac{\log \left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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