Optimal. Leaf size=30 \[ \frac{a^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{b^2 x^4}{4} \]
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Rubi [A] time = 0.0094348, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {43} \[ \frac{a^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{b^2 x^4}{4} \]
Antiderivative was successfully verified.
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Rule 43
Rubi steps
\begin{align*} \int x (a+b x)^2 \, dx &=\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx\\ &=\frac{a^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{b^2 x^4}{4}\\ \end{align*}
Mathematica [A] time = 0.0013115, size = 30, normalized size = 1. \[ \frac{a^2 x^2}{2}+\frac{2}{3} a b x^3+\frac{b^2 x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{2\,ab{x}^{3}}{3}}+{\frac{{b}^{2}{x}^{4}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02898, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, a^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28362, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{4} x^{4} b^{2} + \frac{2}{3} x^{3} b a + \frac{1}{2} x^{2} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.087937, size = 26, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{2}}{2} + \frac{2 a b x^{3}}{3} + \frac{b^{2} x^{4}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21765, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{4} \, b^{2} x^{4} + \frac{2}{3} \, a b x^{3} + \frac{1}{2} \, a^{2} x^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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