Optimal. Leaf size=85 \[ -\frac{27 \left (a+b x^2\right )^{10/3}}{280 a^3 c (c x)^{20/3}}+\frac{9 \left (a+b x^2\right )^{7/3}}{28 a^2 c (c x)^{20/3}}-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{20/3}} \]
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Rubi [A] time = 0.0247993, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {273, 264} \[ -\frac{27 \left (a+b x^2\right )^{10/3}}{280 a^3 c (c x)^{20/3}}+\frac{9 \left (a+b x^2\right )^{7/3}}{28 a^2 c (c x)^{20/3}}-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{20/3}} \]
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)^{23/3}} \, dx &=-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{20/3}}-\frac{3 \int \frac{\left (a+b x^2\right )^{4/3}}{(c x)^{23/3}} \, dx}{2 a}\\ &=-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{20/3}}+\frac{9 \left (a+b x^2\right )^{7/3}}{28 a^2 c (c x)^{20/3}}+\frac{9 \int \frac{\left (a+b x^2\right )^{7/3}}{(c x)^{23/3}} \, dx}{14 a^2}\\ &=-\frac{3 \left (a+b x^2\right )^{4/3}}{8 a c (c x)^{20/3}}+\frac{9 \left (a+b x^2\right )^{7/3}}{28 a^2 c (c x)^{20/3}}-\frac{27 \left (a+b x^2\right )^{10/3}}{280 a^3 c (c x)^{20/3}}\\ \end{align*}
Mathematica [A] time = 0.0180083, size = 52, normalized size = 0.61 \[ -\frac{3 \sqrt [3]{c x} \left (a+b x^2\right )^{4/3} \left (14 a^2-12 a b x^2+9 b^2 x^4\right )}{280 a^3 c^8 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 42, normalized size = 0.5 \begin{align*} -{\frac{3\,x \left ( 9\,{b}^{2}{x}^{4}-12\,ab{x}^{2}+14\,{a}^{2} \right ) }{280\,{a}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}} \left ( cx \right ) ^{-{\frac{23}{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{23}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76081, size = 136, normalized size = 1.6 \begin{align*} -\frac{3 \,{\left (9 \, b^{3} x^{6} - 3 \, a b^{2} x^{4} + 2 \, a^{2} b x^{2} + 14 \, a^{3}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}}{280 \, a^{3} c^{8} x^{7}} \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{23}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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