Optimal. Leaf size=38 \[ \frac{\left (a+b x^3\right )^{7/3}}{7 b^2}-\frac{a \left (a+b x^3\right )^{4/3}}{4 b^2} \]
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Rubi [A] time = 0.0222906, antiderivative size = 38, 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} \[ \frac{\left (a+b x^3\right )^{7/3}}{7 b^2}-\frac{a \left (a+b x^3\right )^{4/3}}{4 b^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int x^5 \sqrt [3]{a+b x^3} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x \sqrt [3]{a+b x} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (-\frac{a \sqrt [3]{a+b x}}{b}+\frac{(a+b x)^{4/3}}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac{a \left (a+b x^3\right )^{4/3}}{4 b^2}+\frac{\left (a+b x^3\right )^{7/3}}{7 b^2}\\ \end{align*}
Mathematica [A] time = 0.0128967, size = 28, normalized size = 0.74 \[ \frac{\left (a+b x^3\right )^{4/3} \left (4 b x^3-3 a\right )}{28 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 25, normalized size = 0.7 \begin{align*} -{\frac{-4\,b{x}^{3}+3\,a}{28\,{b}^{2}} \left ( b{x}^{3}+a \right ) ^{{\frac{4}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03449, size = 41, normalized size = 1.08 \begin{align*} \frac{{\left (b x^{3} + a\right )}^{\frac{7}{3}}}{7 \, b^{2}} - \frac{{\left (b x^{3} + a\right )}^{\frac{4}{3}} a}{4 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66456, size = 78, normalized size = 2.05 \begin{align*} \frac{{\left (4 \, b^{2} x^{6} + a b x^{3} - 3 \, a^{2}\right )}{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{28 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.01663, size = 63, normalized size = 1.66 \begin{align*} \begin{cases} - \frac{3 a^{2} \sqrt [3]{a + b x^{3}}}{28 b^{2}} + \frac{a x^{3} \sqrt [3]{a + b x^{3}}}{28 b} + \frac{x^{6} \sqrt [3]{a + b x^{3}}}{7} & \text{for}\: b \neq 0 \\\frac{\sqrt [3]{a} x^{6}}{6} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11871, size = 39, normalized size = 1.03 \begin{align*} \frac{4 \,{\left (b x^{3} + a\right )}^{\frac{7}{3}} - 7 \,{\left (b x^{3} + a\right )}^{\frac{4}{3}} a}{28 \, b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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