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 = 0.02, antiderivative size = 85, normalized size of antiderivative = 1.00, 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 264
Rule 273
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*}
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Mathematica [A] time = 0.02, 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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fricas [A] time = 1.25, size = 57, normalized size = 0.67 \[ -\frac {3 \, {\left (9 \, b^{3} x^{6} - 3 \, a b^{2} x^{4} + 2 \, a^{2} b x^{2} + 14 \, a^{3}\right )} {\left (b x^{2} + a\right )}^{\frac {1}{3}} \left (c x\right )^{\frac {1}{3}}}{280 \, a^{3} c^{8} x^{7}} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}}}{\left (c x\right )^{\frac {23}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 42, normalized size = 0.49 \[ -\frac {3 \left (b \,x^{2}+a \right )^{\frac {4}{3}} \left (9 b^{2} x^{4}-12 a b \,x^{2}+14 a^{2}\right ) x}{280 \left (c x \right )^{\frac {23}{3}} a^{3}} \]
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 \[ \int \frac {{\left (b x^{2} + a\right )}^{\frac {1}{3}}}{\left (c x\right )^{\frac {23}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (b\,x^2+a\right )}^{1/3}}{{\left (c\,x\right )}^{23/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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