Optimal. Leaf size=31 \[ a A x+\frac{B \left (a+c x^2\right )^2}{4 c}+\frac{1}{3} A c x^3 \]
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Rubi [A] time = 0.0067756, 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} \[ a A x+\frac{B \left (a+c x^2\right )^2}{4 c}+\frac{1}{3} A c x^3 \]
Antiderivative was successfully verified.
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Rule 641
Rubi steps
\begin{align*} \int (A+B x) \left (a+c x^2\right ) \, dx &=\frac{B \left (a+c x^2\right )^2}{4 c}+A \int \left (a+c x^2\right ) \, dx\\ &=a A x+\frac{1}{3} A c x^3+\frac{B \left (a+c x^2\right )^2}{4 c}\\ \end{align*}
Mathematica [A] time = 0.0017835, size = 32, normalized size = 1.03 \[ a A x+\frac{1}{2} a B x^2+\frac{1}{3} A c x^3+\frac{1}{4} B c x^4 \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 27, normalized size = 0.9 \begin{align*}{\frac{Bc{x}^{4}}{4}}+{\frac{Ac{x}^{3}}{3}}+{\frac{aB{x}^{2}}{2}}+aAx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06404, size = 35, normalized size = 1.13 \begin{align*} \frac{1}{4} \, B c x^{4} + \frac{1}{3} \, A c x^{3} + \frac{1}{2} \, B a x^{2} + A a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50926, size = 66, normalized size = 2.13 \begin{align*} \frac{1}{4} x^{4} c B + \frac{1}{3} x^{3} c A + \frac{1}{2} x^{2} a B + x a A \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.126366, size = 29, normalized size = 0.94 \begin{align*} A a x + \frac{A c x^{3}}{3} + \frac{B a x^{2}}{2} + \frac{B c x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28257, size = 35, normalized size = 1.13 \begin{align*} \frac{1}{4} \, B c x^{4} + \frac{1}{3} \, A c x^{3} + \frac{1}{2} \, B a x^{2} + A a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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