Optimal. Leaf size=30 \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]
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Rubi [A] time = 0.0195837, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1584, 266, 43} \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 266
Rule 43
Rubi steps
\begin{align*} \int x \left (a x+b x^3\right )^2 \, dx &=\int x^3 \left (a+b x^2\right )^2 \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^2 \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a^2 x+2 a b x^2+b^2 x^3\right ) \, dx,x,x^2\right )\\ &=\frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8}\\ \end{align*}
Mathematica [A] time = 0.0009825, size = 30, normalized size = 1. \[ \frac{a^2 x^4}{4}+\frac{1}{3} a b x^6+\frac{b^2 x^8}{8} \]
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}^{4}}{4}}+{\frac{ab{x}^{6}}{3}}+{\frac{{b}^{2}{x}^{8}}{8}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02882, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{8} \, b^{2} x^{8} + \frac{1}{3} \, a b x^{6} + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.1766, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{8} x^{8} b^{2} + \frac{1}{3} x^{6} b a + \frac{1}{4} x^{4} a^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.059262, size = 24, normalized size = 0.8 \begin{align*} \frac{a^{2} x^{4}}{4} + \frac{a b x^{6}}{3} + \frac{b^{2} x^{8}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16584, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{8} \, b^{2} x^{8} + \frac{1}{3} \, a b x^{6} + \frac{1}{4} \, a^{2} x^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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