Optimal. Leaf size=56 \[ \frac{a^2 \sqrt [3]{a+b x^3}}{b^3}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^3}-\frac{a \left (a+b x^3\right )^{4/3}}{2 b^3} \]
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Rubi [A] time = 0.0330715, antiderivative size = 56, 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} \[ \frac{a^2 \sqrt [3]{a+b x^3}}{b^3}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^3}-\frac{a \left (a+b x^3\right )^{4/3}}{2 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^8}{\left (a+b x^3\right )^{2/3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{(a+b x)^{2/3}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 (a+b x)^{2/3}}-\frac{2 a \sqrt [3]{a+b x}}{b^2}+\frac{(a+b x)^{4/3}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac{a^2 \sqrt [3]{a+b x^3}}{b^3}-\frac{a \left (a+b x^3\right )^{4/3}}{2 b^3}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^3}\\ \end{align*}
Mathematica [A] time = 0.016996, size = 39, normalized size = 0.7 \[ \frac{\sqrt [3]{a+b x^3} \left (9 a^2-3 a b x^3+2 b^2 x^6\right )}{14 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 36, normalized size = 0.6 \begin{align*}{\frac{2\,{b}^{2}{x}^{6}-3\,{x}^{3}ab+9\,{a}^{2}}{14\,{b}^{3}}\sqrt [3]{b{x}^{3}+a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.993589, size = 62, normalized size = 1.11 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{7}{3}}}{7 \, b^{3}} - \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} a}{2 \, b^{3}} + \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{2}}{b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.43133, size = 81, normalized size = 1.45 \begin{align*} \frac{{\left (2 \, b^{2} x^{6} - 3 \, a b x^{3} + 9 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{14 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.09424, size = 68, normalized size = 1.21 \begin{align*} \begin{cases} \frac{9 a^{2} \sqrt [3]{a + b x^{3}}}{14 b^{3}} - \frac{3 a x^{3} \sqrt [3]{a + b x^{3}}}{14 b^{2}} + \frac{x^{6} \sqrt [3]{a + b x^{3}}}{7 b} & \text{for}\: b \neq 0 \\\frac{x^{9}}{9 a^{\frac{2}{3}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11753, size = 58, normalized size = 1.04 \begin{align*} \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a + 14 \,{\left (b x^{3} + a\right )}^{\frac{1}{3}} a^{2}}{14 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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