Optimal. Leaf size=448 \[ -\frac{8 a^2 \sqrt [3]{c} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} \text{EllipticF}\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right ),\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{135 \sqrt [4]{3} b \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac{(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}+\frac{8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c} \]
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Rubi [A] time = 0.745642, antiderivative size = 448, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {279, 321, 329, 241, 225} \[ -\frac{8 a^2 \sqrt [3]{c} \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt{\frac{\frac{b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac{\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac{c^{2/3}-\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{135 \sqrt [4]{3} b \sqrt{-\frac{\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac{\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}+\frac{16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac{(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}+\frac{8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 329
Rule 241
Rule 225
Rubi steps
\begin{align*} \int (c x)^{4/3} \left (a+b x^2\right )^{4/3} \, dx &=\frac{(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}+\frac{1}{15} (8 a) \int (c x)^{4/3} \sqrt [3]{a+b x^2} \, dx\\ &=\frac{8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c}+\frac{(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}+\frac{1}{135} \left (16 a^2\right ) \int \frac{(c x)^{4/3}}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac{16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac{8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c}+\frac{(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}-\frac{\left (16 a^3 c^2\right ) \int \frac{1}{(c x)^{2/3} \left (a+b x^2\right )^{2/3}} \, dx}{405 b}\\ &=\frac{16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac{8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c}+\frac{(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}-\frac{\left (16 a^3 c\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 )}{135 b}\\ &=\frac{16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac{8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c}+\frac{(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}-\frac{\left (16 a^3 c\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 )}{135 b \sqrt{\frac{a}{a+b x^2}} \sqrt{a+b x^2}}\\ &=\frac{16 a^2 c \sqrt [3]{c x} \sqrt [3]{a+b x^2}}{135 b}+\frac{8 a (c x)^{7/3} \sqrt [3]{a+b x^2}}{45 c}+\frac{(c x)^{7/3} \left (a+b x^2\right )^{4/3}}{5 c}-\frac{8 a^2 \sqrt [3]{c} \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 )}{135 \sqrt [4]{3} b \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 = 0.0477848, size = 89, normalized size = 0.2 \[ \frac{c \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (\left (a+b x^2\right )^2 \sqrt [3]{\frac{b x^2}{a}+1}-a^2 \, _2F_1\left (-\frac{4}{3},\frac{1}{6};\frac{7}{6};-\frac{b x^2}{a}\right )\right )}{5 b \sqrt [3]{\frac{b x^2}{a}+1}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int \left ( cx \right ) ^{{\frac{4}{3}}} \left ( b{x}^{2}+a \right ) ^{{\frac{4}{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{\left (b x^{2} + a\right )}^{\frac{4}{3}} \left (c x\right )^{\frac{4}{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 ({\left (b c x^{3} + a c x\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}} \left (c x\right )^{\frac{1}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 64.5754, size = 46, normalized size = 0.1 \begin{align*} \frac{a^{\frac{4}{3}} c^{\frac{4}{3}} x^{\frac{7}{3}} \Gamma \left (\frac{7}{6}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{7}{6} \\ \frac{13}{6} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac{13}{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{\left (b x^{2} + a\right )}^{\frac{4}{3}} \left (c x\right )^{\frac{4}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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