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 = 0.02, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 264
Rule 273
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*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 0.60 \[ -\frac {3 x \left (a-3 b x^2\right ) \sqrt [3]{a+b x^2}}{8 a^2 (c x)^{11/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.14, size = 35, normalized size = 0.61 \[ \frac {3 \, {\left (3 \, b x^{2} - a\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {1}{3}}}{8 \, a^{2} c^{4} x^{3}} \]
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 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {2}{3}} \left (c x\right )^{\frac {11}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.51 \[ -\frac {3 \left (b \,x^{2}+a \right )^{\frac {1}{3}} \left (-3 b \,x^{2}+a \right ) x}{8 \left (c x \right )^{\frac {11}{3}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {2}{3}} \left (c x\right )^{\frac {11}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (c\,x\right )}^{11/3}\,{\left (b\,x^2+a\right )}^{2/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 109.45, size = 78, normalized size = 1.37 \[ - \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {4}{3}\right )}{6 a c^{\frac {11}{3}} x^{2} \Gamma \left (\frac {2}{3}\right )} + \frac {b^{\frac {4}{3}} \sqrt [3]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {4}{3}\right )}{2 a^{2} c^{\frac {11}{3}} \Gamma \left (\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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