Optimal. Leaf size=52 \[ \frac {\left (a x^3+b x^6\right )^{8/3}}{11 b x^5}-\frac {3 a \left (a x^3+b x^6\right )^{8/3}}{88 b^2 x^8} \]
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Rubi [A] time = 0.05, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2002, 2014} \begin {gather*} \frac {\left (a x^3+b x^6\right )^{8/3}}{11 b x^5}-\frac {3 a \left (a x^3+b x^6\right )^{8/3}}{88 b^2 x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rubi steps
\begin {align*} \int \left (a x^3+b x^6\right )^{5/3} \, dx &=\frac {\left (a x^3+b x^6\right )^{8/3}}{11 b x^5}-\frac {(3 a) \int \frac {\left (a x^3+b x^6\right )^{5/3}}{x^3} \, dx}{11 b}\\ &=-\frac {3 a \left (a x^3+b x^6\right )^{8/3}}{88 b^2 x^8}+\frac {\left (a x^3+b x^6\right )^{8/3}}{11 b x^5}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 42, normalized size = 0.81 \begin {gather*} \frac {x \left (a+b x^3\right )^3 \left (8 b x^3-3 a\right )}{88 b^2 \sqrt [3]{x^3 \left (a+b x^3\right )}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.28, size = 57, normalized size = 1.10 \begin {gather*} \frac {\left (a x^3+b x^6\right )^{2/3} \left (-3 a^3+2 a^2 b x^3+13 a b^2 x^6+8 b^3 x^9\right )}{88 b^2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.97, size = 53, normalized size = 1.02 \begin {gather*} \frac {{\left (8 \, b^{3} x^{9} + 13 \, a b^{2} x^{6} + 2 \, a^{2} b x^{3} - 3 \, a^{3}\right )} {\left (b x^{6} + a x^{3}\right )}^{\frac {2}{3}}}{88 \, b^{2} x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (b x^{6} + a x^{3}\right )}^{\frac {5}{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 39, normalized size = 0.75 \begin {gather*} -\frac {\left (b \,x^{3}+a \right ) \left (-8 b \,x^{3}+3 a \right ) \left (b \,x^{6}+a \,x^{3}\right )^{\frac {5}{3}}}{88 b^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 46, normalized size = 0.88 \begin {gather*} \frac {{\left (8 \, b^{3} x^{9} + 13 \, a b^{2} x^{6} + 2 \, a^{2} b x^{3} - 3 \, a^{3}\right )} {\left (b x^{3} + a\right )}^{\frac {2}{3}}}{88 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 40, normalized size = 0.77 \begin {gather*} -\frac {{\left (b\,x^3+a\right )}^2\,{\left (b\,x^6+a\,x^3\right )}^{2/3}\,\left (3\,a-8\,b\,x^3\right )}{88\,b^2\,x^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a x^{3} + b x^{6}\right )^{\frac {5}{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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