Optimal. Leaf size=38 \[ \frac {\left (a+b x^3\right )^{5/3}}{5 b^2}-\frac {a \left (a+b x^3\right )^{2/3}}{2 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, 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} \[ \frac {\left (a+b x^3\right )^{5/3}}{5 b^2}-\frac {a \left (a+b x^3\right )^{2/3}}{2 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt [3]{a+b x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{a+b x}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (-\frac {a}{b \sqrt [3]{a+b x}}+\frac {(a+b x)^{2/3}}{b}\right ) \, dx,x,x^3\right )\\ &=-\frac {a \left (a+b x^3\right )^{2/3}}{2 b^2}+\frac {\left (a+b x^3\right )^{5/3}}{5 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 28, normalized size = 0.74 \[ \frac {\left (a+b x^3\right )^{2/3} \left (2 b x^3-3 a\right )}{10 b^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 24, normalized size = 0.63 \[ \frac {{\left (2 \, b x^{3} - 3 \, a\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{10 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 30, normalized size = 0.79 \[ \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{5 \, b^{2}} - \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 25, normalized size = 0.66 \[ -\frac {\left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (-2 b \,x^{3}+3 a \right )}{10 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 30, normalized size = 0.79 \[ \frac {{\left (b x^{3} + a\right )}^{\frac {5}{3}}}{5 \, b^{2}} - \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}} a}{2 \, b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 24, normalized size = 0.63 \[ -\frac {{\left (b\,x^3+a\right )}^{2/3}\,\left (3\,a-2\,b\,x^3\right )}{10\,b^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.40, size = 44, normalized size = 1.16 \[ \begin {cases} - \frac {3 a \left (a + b x^{3}\right )^{\frac {2}{3}}}{10 b^{2}} + \frac {x^{3} \left (a + b x^{3}\right )^{\frac {2}{3}}}{5 b} & \text {for}\: b \neq 0 \\\frac {x^{6}}{6 \sqrt [3]{a}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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