Optimal. Leaf size=28 \[ -\frac {3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{8/3}} \]
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Rubi [A] time = 0.01, antiderivative size = 28, normalized size of antiderivative = 1.00, 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} \[ -\frac {3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{8/3}} \]
Antiderivative was successfully verified.
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Rule 264
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{11/3}} \, dx &=-\frac {3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{8/3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 0.93 \[ -\frac {3 x \left (a+b x^2\right )^{4/3}}{8 a (c x)^{11/3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 25, normalized size = 0.89 \[ -\frac {3 \, {\left (b x^{2} + a\right )}^{\frac {4}{3}} \left (c x\right )^{\frac {1}{3}}}{8 \, a c^{4} x^{3}} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}}}{\left (c x\right )^{\frac {11}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.75 \[ -\frac {3 \left (b \,x^{2}+a \right )^{\frac {4}{3}} x}{8 \left (c x \right )^{\frac {11}{3}} a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.45, size = 35, normalized size = 1.25 \[ -\frac {3 \, {\left (b c^{\frac {1}{3}} x^{3} + a c^{\frac {1}{3}} x\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}}}{8 \, a c^{4} x^{\frac {11}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (b\,x^2+a\right )}^{1/3}}{{\left (c\,x\right )}^{11/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 56.59, size = 78, normalized size = 2.79 \[ \frac {\sqrt [3]{b} \sqrt [3]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {4}{3}\right )}{2 c^{\frac {11}{3}} x^{2} \Gamma \left (- \frac {1}{3}\right )} + \frac {b^{\frac {4}{3}} \sqrt [3]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {4}{3}\right )}{2 a c^{\frac {11}{3}} \Gamma \left (- \frac {1}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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