Optimal. Leaf size=31 \[ \frac{e \left (a+c x^2\right )^2}{4 c}+a d x+\frac{1}{3} c d x^3 \]
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Rubi [A] time = 0.006554, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {641} \[ \frac{e \left (a+c x^2\right )^2}{4 c}+a d x+\frac{1}{3} c d x^3 \]
Antiderivative was successfully verified.
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Rule 641
Rubi steps
\begin{align*} \int (d+e x) \left (a+c x^2\right ) \, dx &=\frac{e \left (a+c x^2\right )^2}{4 c}+d \int \left (a+c x^2\right ) \, dx\\ &=a d x+\frac{1}{3} c d x^3+\frac{e \left (a+c x^2\right )^2}{4 c}\\ \end{align*}
Mathematica [A] time = 0.0013999, size = 32, normalized size = 1.03 \[ a d x+\frac{1}{2} a e x^2+\frac{1}{3} c d x^3+\frac{1}{4} c e x^4 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 27, normalized size = 0.9 \begin{align*}{\frac{ce{x}^{4}}{4}}+{\frac{cd{x}^{3}}{3}}+{\frac{ae{x}^{2}}{2}}+adx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12008, size = 35, normalized size = 1.13 \begin{align*} \frac{1}{4} \, c e x^{4} + \frac{1}{3} \, c d x^{3} + \frac{1}{2} \, a e x^{2} + a d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6751, size = 66, normalized size = 2.13 \begin{align*} \frac{1}{4} x^{4} e c + \frac{1}{3} x^{3} d c + \frac{1}{2} x^{2} e a + x d a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.069461, size = 29, normalized size = 0.94 \begin{align*} a d x + \frac{a e x^{2}}{2} + \frac{c d x^{3}}{3} + \frac{c e x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.26689, size = 38, normalized size = 1.23 \begin{align*} \frac{1}{4} \, c x^{4} e + \frac{1}{3} \, c d x^{3} + \frac{1}{2} \, a x^{2} e + a d x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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