Optimal. Leaf size=113 \[ \frac{243 \left (a+b x^2\right )^{10/3}}{280 a^4 c (c x)^{20/3}}-\frac{81 \left (a+b x^2\right )^{7/3}}{28 a^3 c (c x)^{20/3}}+\frac{27 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{20/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}} \]
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Rubi [A] time = 0.0393894, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {273, 264} \[ \frac{243 \left (a+b x^2\right )^{10/3}}{280 a^4 c (c x)^{20/3}}-\frac{81 \left (a+b x^2\right )^{7/3}}{28 a^3 c (c x)^{20/3}}+\frac{27 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{20/3}}-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}} \]
Antiderivative was successfully verified.
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Rule 273
Rule 264
Rubi steps
\begin{align*} \int \frac{1}{(c x)^{23/3} \left (a+b x^2\right )^{2/3}} \, dx &=-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}}-\frac{9 \int \frac{\sqrt [3]{a+b x^2}}{(c x)^{23/3}} \, dx}{a}\\ &=-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}}+\frac{27 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{20/3}}+\frac{27 \int \frac{\left (a+b x^2\right )^{4/3}}{(c x)^{23/3}} \, dx}{2 a^2}\\ &=-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}}+\frac{27 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{20/3}}-\frac{81 \left (a+b x^2\right )^{7/3}}{28 a^3 c (c x)^{20/3}}-\frac{81 \int \frac{\left (a+b x^2\right )^{7/3}}{(c x)^{23/3}} \, dx}{14 a^3}\\ &=-\frac{3 \sqrt [3]{a+b x^2}}{2 a c (c x)^{20/3}}+\frac{27 \left (a+b x^2\right )^{4/3}}{8 a^2 c (c x)^{20/3}}-\frac{81 \left (a+b x^2\right )^{7/3}}{28 a^3 c (c x)^{20/3}}+\frac{243 \left (a+b x^2\right )^{10/3}}{280 a^4 c (c x)^{20/3}}\\ \end{align*}
Mathematica [A] time = 0.0291058, size = 63, normalized size = 0.56 \[ \frac{3 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (18 a^2 b x^2-14 a^3-27 a b^2 x^4+81 b^3 x^6\right )}{280 a^4 c^8 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 53, normalized size = 0.5 \begin{align*} -{\frac{3\,x \left ( -81\,{b}^{3}{x}^{6}+27\,a{b}^{2}{x}^{4}-18\,{a}^{2}b{x}^{2}+14\,{a}^{3} \right ) }{280\,{a}^{4}}\sqrt [3]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{23}{3}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.38595, size = 86, normalized size = 0.76 \begin{align*} \frac{3 \,{\left (81 \, b^{4} x^{9} + 54 \, a b^{3} x^{7} - 9 \, a^{2} b^{2} x^{5} + 4 \, a^{3} b x^{3} - 14 \, a^{4} x\right )}}{280 \,{\left (b x^{2} + a\right )}^{\frac{2}{3}} a^{4} c^{\frac{23}{3}} x^{\frac{23}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.54331, size = 139, normalized size = 1.23 \begin{align*} \frac{3 \,{\left (81 \, b^{3} x^{6} - 27 \, a b^{2} x^{4} + 18 \, 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^{4} 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{1}{{\left (b x^{2} + a\right )}^{\frac{2}{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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