Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^3}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^3}-\frac{a \left (a+b x^3\right )^{8/3}}{4 b^3} \]
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Rubi [A] time = 0.0343528, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^3}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^3}-\frac{a \left (a+b x^3\right )^{8/3}}{4 b^3} \]
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 )^{2/3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^2 (a+b x)^{2/3} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2 (a+b x)^{2/3}}{b^2}-\frac{2 a (a+b x)^{5/3}}{b^2}+\frac{(a+b x)^{8/3}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac{a^2 \left (a+b x^3\right )^{5/3}}{5 b^3}-\frac{a \left (a+b x^3\right )^{8/3}}{4 b^3}+\frac{\left (a+b x^3\right )^{11/3}}{11 b^3}\\ \end{align*}
Mathematica [A] time = 0.0182656, size = 39, normalized size = 0.66 \[ \frac{\left (a+b x^3\right )^{5/3} \left (9 a^2-15 a b x^3+20 b^2 x^6\right )}{220 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 36, normalized size = 0.6 \begin{align*}{\frac{20\,{b}^{2}{x}^{6}-15\,{x}^{3}ab+9\,{a}^{2}}{220\,{b}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{5}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03002, size = 63, normalized size = 1.07 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{11}{3}}}{11 \, b^{3}} - \frac{{\left (b x^{3} + a\right )}^{\frac{8}{3}} a}{4 \, b^{3}} + \frac{{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{2}}{5 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.5583, size = 105, normalized size = 1.78 \begin{align*} \frac{{\left (20 \, b^{3} x^{9} + 5 \, a b^{2} x^{6} - 6 \, a^{2} b x^{3} + 9 \, a^{3}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{220 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.05843, size = 87, normalized size = 1.47 \begin{align*} \begin{cases} \frac{9 a^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{220 b^{3}} - \frac{3 a^{2} x^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{110 b^{2}} + \frac{a x^{6} \left (a + b x^{3}\right )^{\frac{2}{3}}}{44 b} + \frac{x^{9} \left (a + b x^{3}\right )^{\frac{2}{3}}}{11} & \text{for}\: b \neq 0 \\\frac{a^{\frac{2}{3}} x^{9}}{9} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10149, size = 58, normalized size = 0.98 \begin{align*} \frac{20 \,{\left (b x^{3} + a\right )}^{\frac{11}{3}} - 55 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} a + 44 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a^{2}}{220 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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