Optimal. Leaf size=43 \[ \frac{1}{4} a^2 b x^{12}+\frac{a^3 x^9}{9}+\frac{1}{5} a b^2 x^{15}+\frac{b^3 x^{18}}{18} \]
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Rubi [A] time = 0.024537, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac{1}{4} a^2 b x^{12}+\frac{a^3 x^9}{9}+\frac{1}{5} a b^2 x^{15}+\frac{b^3 x^{18}}{18} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^8 \left (a+b x^3\right )^3 \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 (a+b x)^3 \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (a^3 x^2+3 a^2 b x^3+3 a b^2 x^4+b^3 x^5\right ) \, dx,x,x^3\right )\\ &=\frac{a^3 x^9}{9}+\frac{1}{4} a^2 b x^{12}+\frac{1}{5} a b^2 x^{15}+\frac{b^3 x^{18}}{18}\\ \end{align*}
Mathematica [A] time = 0.001715, size = 43, normalized size = 1. \[ \frac{1}{4} a^2 b x^{12}+\frac{a^3 x^9}{9}+\frac{1}{5} a b^2 x^{15}+\frac{b^3 x^{18}}{18} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 36, normalized size = 0.8 \begin{align*}{\frac{{a}^{3}{x}^{9}}{9}}+{\frac{{a}^{2}b{x}^{12}}{4}}+{\frac{a{b}^{2}{x}^{15}}{5}}+{\frac{{b}^{3}{x}^{18}}{18}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.953135, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{18} \, b^{3} x^{18} + \frac{1}{5} \, a b^{2} x^{15} + \frac{1}{4} \, a^{2} b x^{12} + \frac{1}{9} \, a^{3} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.44471, size = 85, normalized size = 1.98 \begin{align*} \frac{1}{18} x^{18} b^{3} + \frac{1}{5} x^{15} b^{2} a + \frac{1}{4} x^{12} b a^{2} + \frac{1}{9} x^{9} a^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.084648, size = 36, normalized size = 0.84 \begin{align*} \frac{a^{3} x^{9}}{9} + \frac{a^{2} b x^{12}}{4} + \frac{a b^{2} x^{15}}{5} + \frac{b^{3} x^{18}}{18} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11864, size = 47, normalized size = 1.09 \begin{align*} \frac{1}{18} \, b^{3} x^{18} + \frac{1}{5} \, a b^{2} x^{15} + \frac{1}{4} \, a^{2} b x^{12} + \frac{1}{9} \, a^{3} x^{9} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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