Optimal. Leaf size=414 \[ \frac{8 a \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} \text{EllipticF}\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{9 \sqrt [4]{3} c^{5/3} \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{\sqrt [3]{c x} \left (a+b x^2\right )^{4/3}}{3 c}+\frac{8 a \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{9 c} \]
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Rubi [A] time = 0.694529, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {279, 329, 241, 225} \[ \frac{8 a \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{9 \sqrt [4]{3} c^{5/3} \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{\sqrt [3]{c x} \left (a+b x^2\right )^{4/3}}{3 c}+\frac{8 a \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{9 c} \]
Antiderivative was successfully verified.
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Rule 279
Rule 329
Rule 241
Rule 225
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^{4/3}}{(c x)^{2/3}} \, dx &=\frac{\sqrt [3]{c x} \left (a+b x^2\right )^{4/3}}{3 c}+\frac{1}{9} (8 a) \int \frac{\sqrt [3]{a+b x^2}}{(c x)^{2/3}} \, dx\\ &=\frac{8 a \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{9 c}+\frac{\sqrt [3]{c x} \left (a+b x^2\right )^{4/3}}{3 c}+\frac{1}{27} \left (16 a^2\right ) \int \frac{1}{(c x)^{2/3} \left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac{8 a \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{9 c}+\frac{\sqrt [3]{c x} \left (a+b x^2\right )^{4/3}}{3 c}+\frac{\left (16 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (a+\frac{b x^6}{c^2}\right )^{2/3}} \, dx,x,\sqrt [3]{c x}\right )}{9 c}\\ &=\frac{8 a \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{9 c}+\frac{\sqrt [3]{c x} \left (a+b x^2\right )^{4/3}}{3 c}+\frac{\left (16 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{b x^6}{c^2}}} \, dx,x,\frac{\sqrt [3]{c x}}{\sqrt [6]{a+b x^2}}\right )}{9 c \sqrt{\frac{a}{a+b x^2}} \sqrt{a+b x^2}}\\ &=\frac{8 a \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{9 c}+\frac{\sqrt [3]{c x} \left (a+b x^2\right )^{4/3}}{3 c}+\frac{8 a \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{c^{4/3}+\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{9 \sqrt [4]{3} c^{5/3} \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 0.0107466, size = 55, normalized size = 0.13 \[ \frac{3 a x \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac{4}{3},\frac{1}{6};\frac{7}{6};-\frac{b x^2}{a}\right )}{(c x)^{2/3} \sqrt [3]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.014, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+a \right ) ^{{\frac{4}{3}}} \left ( cx \right ) ^{-{\frac{2}{3}}}}\, dx \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{4}{3}}}{\left (c x\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{4}{3}} \left (c x\right )^{\frac{1}{3}}}{c x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 14.3585, size = 46, normalized size = 0.11 \begin{align*} \frac{a^{\frac{4}{3}} \sqrt [3]{x} \Gamma \left (\frac{1}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{6} \\ \frac{7}{6} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 c^{\frac{2}{3}} \Gamma \left (\frac{7}{6}\right )} \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{4}{3}}}{\left (c x\right )^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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