Optimal. Leaf size=57 \[ \frac{9 \left (a+b x^2\right )^{7/3}}{56 a^2 c (c x)^{14/3}}-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{14/3}} \]
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Rubi [A] time = 0.0147292, 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} \[ \frac{9 \left (a+b x^2\right )^{7/3}}{56 a^2 c (c x)^{14/3}}-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{14/3}} \]
Antiderivative was successfully verified.
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Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a+b x^2}}{(c x)^{17/3}} \, dx &=-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{14/3}}-\frac{3 \int \frac{\left (a+b x^2\right )^{4/3}}{(c x)^{17/3}} \, dx}{4 a}\\ &=-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{14/3}}+\frac{9 \left (a+b x^2\right )^{7/3}}{56 a^2 c (c x)^{14/3}}\\ \end{align*}
Mathematica [A] time = 0.0169664, size = 41, normalized size = 0.72 \[ \frac{3 \sqrt [3]{c x} \left (a+b x^2\right )^{4/3} \left (3 b x^2-4 a\right )}{56 a^2 c^6 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 31, normalized size = 0.5 \begin{align*} -{\frac{3\,x \left ( -3\,b{x}^{2}+4\,a \right ) }{56\,{a}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}} \left ( cx \right ) ^{-{\frac{17}{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{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{\left (c x\right )^{\frac{17}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49901, size = 108, normalized size = 1.89 \begin{align*} \frac{3 \,{\left (3 \, b^{2} x^{4} - a b x^{2} - 4 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}}{56 \, a^{2} c^{6} x^{5}} \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{{\left (b x^{2} + a\right )}^{\frac{1}{3}}}{\left (c x\right )^{\frac{17}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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