Optimal. Leaf size=57 \[ \frac{9 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{8/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{8/3}} \]
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Rubi [A] time = 0.0153678, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {273, 264} \[ \frac{9 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{8/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{8/3}} \]
Antiderivative was successfully verified.
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Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{11/3} \left (a+b x^2\right )^{2/3}} \, dx &=-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{8/3}}-\frac{3 \int \frac{\sqrt [3]{a+b x^2}}{(c x)^{11/3}} \, dx}{a}\\ &=-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{8/3}}+\frac{9 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{8/3}}\\ \end{align*}
Mathematica [A] time = 0.0180092, size = 34, normalized size = 0.6 \[ -\frac{3 x \left (a-3 b x^2\right ) \sqrt [3]{a+b x^2}}{8 a^2 (c x)^{11/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 0.5 \begin{align*} -{\frac{3\,x \left ( -3\,b{x}^{2}+a \right ) }{8\,{a}^{2}}\sqrt [3]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{11}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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{11}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.04231, size = 85, normalized size = 1.49 \begin{align*} \frac{3 \,{\left (3 \, b x^{2} - a\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}}{8 \, a^{2} c^{4} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{11}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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