Optimal. Leaf size=14 \[ a e x+\frac{1}{2} c d x^2 \]
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Rubi [A] time = 0.0079364, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {24} \[ a e x+\frac{1}{2} c d x^2 \]
Antiderivative was successfully verified.
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Rule 24
Rubi steps
\begin{align*} \int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{d+e x} \, dx &=\frac{\int \left (a e^3+c d e^2 x\right ) \, dx}{e^2}\\ &=a e x+\frac{1}{2} c d x^2\\ \end{align*}
Mathematica [A] time = 0.0008907, size = 14, normalized size = 1. \[ a e x+\frac{1}{2} c d x^2 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 13, normalized size = 0.9 \begin{align*} aex+{\frac{cd{x}^{2}}{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03206, size = 16, normalized size = 1.14 \begin{align*} \frac{1}{2} \, c d x^{2} + a e x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.59621, size = 28, normalized size = 2. \begin{align*} \frac{1}{2} \, c d x^{2} + a e x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.128321, size = 12, normalized size = 0.86 \begin{align*} a e x + \frac{c d x^{2}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15146, size = 26, normalized size = 1.86 \begin{align*} \frac{1}{2} \,{\left (c d x^{2} e^{2} + 2 \, a x e^{3}\right )} e^{\left (-2\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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