Optimal. Leaf size=28 \[ -\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{2/3}} \]
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Rubi [A] time = 0.0062157, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {264} \[ -\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{2/3}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{5/3} \left (a+b x^2\right )^{2/3}} \, dx &=-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0055331, size = 26, normalized size = 0.93 \[ -\frac{3 x \sqrt [3]{a+b x^2}}{2 a (c x)^{5/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 21, normalized size = 0.8 \begin{align*} -{\frac{3\,x}{2\,a}\sqrt [3]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{5}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.29757, size = 47, normalized size = 1.68 \begin{align*} -\frac{3 \,{\left (b c^{\frac{1}{3}} x^{3} + a c^{\frac{1}{3}} x\right )}}{2 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a c^{2} x^{\frac{5}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01498, size = 62, normalized size = 2.21 \begin{align*} -\frac{3 \,{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}}{2 \, a c^{2} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 9.03116, size = 36, normalized size = 1.29 \begin{align*} \frac{\sqrt [3]{b} \sqrt [3]{\frac{a}{b x^{2}} + 1} \Gamma \left (- \frac{1}{3}\right )}{2 a c^{\frac{5}{3}} \Gamma \left (\frac{2}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{2}{3}} \left (c x\right )^{\frac{5}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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