Optimal. Leaf size=59 \[ \frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^3}+\frac{\left (a+b x^3\right )^{8/3}}{8 b^3}-\frac{2 a \left (a+b x^3\right )^{5/3}}{5 b^3} \]
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Rubi [A] time = 0.0349255, 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} \[ \frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^3}+\frac{\left (a+b x^3\right )^{8/3}}{8 b^3}-\frac{2 a \left (a+b x^3\right )^{5/3}}{5 b^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^8}{\sqrt [3]{a+b x^3}} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x^2}{\sqrt [3]{a+b x}} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{a^2}{b^2 \sqrt [3]{a+b x}}-\frac{2 a (a+b x)^{2/3}}{b^2}+\frac{(a+b x)^{5/3}}{b^2}\right ) \, dx,x,x^3\right )\\ &=\frac{a^2 \left (a+b x^3\right )^{2/3}}{2 b^3}-\frac{2 a \left (a+b x^3\right )^{5/3}}{5 b^3}+\frac{\left (a+b x^3\right )^{8/3}}{8 b^3}\\ \end{align*}
Mathematica [A] time = 0.0179953, size = 39, normalized size = 0.66 \[ \frac{\left (a+b x^3\right )^{2/3} \left (9 a^2-6 a b x^3+5 b^2 x^6\right )}{40 b^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 36, normalized size = 0.6 \begin{align*}{\frac{5\,{b}^{2}{x}^{6}-6\,{x}^{3}ab+9\,{a}^{2}}{40\,{b}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0767, size = 63, normalized size = 1.07 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{8}{3}}}{8 \, b^{3}} - \frac{2 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a}{5 \, b^{3}} + \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{2}}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.60175, size = 81, normalized size = 1.37 \begin{align*} \frac{{\left (5 \, b^{2} x^{6} - 6 \, a b x^{3} + 9 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{40 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.89038, size = 68, normalized size = 1.15 \begin{align*} \begin{cases} \frac{9 a^{2} \left (a + b x^{3}\right )^{\frac{2}{3}}}{40 b^{3}} - \frac{3 a x^{3} \left (a + b x^{3}\right )^{\frac{2}{3}}}{20 b^{2}} + \frac{x^{6} \left (a + b x^{3}\right )^{\frac{2}{3}}}{8 b} & \text{for}\: b \neq 0 \\\frac{x^{9}}{9 \sqrt [3]{a}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12202, size = 58, normalized size = 0.98 \begin{align*} \frac{5 \,{\left (b x^{3} + a\right )}^{\frac{8}{3}} - 16 \,{\left (b x^{3} + a\right )}^{\frac{5}{3}} a + 20 \,{\left (b x^{3} + a\right )}^{\frac{2}{3}} a^{2}}{40 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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