Optimal. Leaf size=311 \[ \frac{48\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right ),4 \sqrt{3}-7\right )}{935 b^2 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{48 a^2 x \sqrt [3]{a+b x^2}}{935 b}+\frac{24}{187} a x^3 \sqrt [3]{a+b x^2}+\frac{3}{17} x^3 \left (a+b x^2\right )^{4/3} \]
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Rubi [A] time = 0.185407, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {279, 321, 236, 219} \[ \frac{48\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{935 b^2 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{48 a^2 x \sqrt [3]{a+b x^2}}{935 b}+\frac{24}{187} a x^3 \sqrt [3]{a+b x^2}+\frac{3}{17} x^3 \left (a+b x^2\right )^{4/3} \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 236
Rule 219
Rubi steps
\begin{align*} \int x^2 \left (a+b x^2\right )^{4/3} \, dx &=\frac{3}{17} x^3 \left (a+b x^2\right )^{4/3}+\frac{1}{17} (8 a) \int x^2 \sqrt [3]{a+b x^2} \, dx\\ &=\frac{24}{187} a x^3 \sqrt [3]{a+b x^2}+\frac{3}{17} x^3 \left (a+b x^2\right )^{4/3}+\frac{1}{187} \left (16 a^2\right ) \int \frac{x^2}{\left (a+b x^2\right )^{2/3}} \, dx\\ &=\frac{48 a^2 x \sqrt [3]{a+b x^2}}{935 b}+\frac{24}{187} a x^3 \sqrt [3]{a+b x^2}+\frac{3}{17} x^3 \left (a+b x^2\right )^{4/3}-\frac{\left (48 a^3\right ) \int \frac{1}{\left (a+b x^2\right )^{2/3}} \, dx}{935 b}\\ &=\frac{48 a^2 x \sqrt [3]{a+b x^2}}{935 b}+\frac{24}{187} a x^3 \sqrt [3]{a+b x^2}+\frac{3}{17} x^3 \left (a+b x^2\right )^{4/3}-\frac{\left (72 a^3 \sqrt{b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a+b x^2}\right )}{935 b^2 x}\\ &=\frac{48 a^2 x \sqrt [3]{a+b x^2}}{935 b}+\frac{24}{187} a x^3 \sqrt [3]{a+b x^2}+\frac{3}{17} x^3 \left (a+b x^2\right )^{4/3}+\frac{48\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}}\right )|-7+4 \sqrt{3}\right )}{935 b^2 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}\\ \end{align*}
Mathematica [C] time = 0.0626881, size = 67, normalized size = 0.22 \[ \frac{3 x \sqrt [3]{a+b x^2} \left (\left (a+b x^2\right )^2-\frac{a^2 \, _2F_1\left (-\frac{4}{3},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )}{\sqrt [3]{\frac{b x^2}{a}+1}}\right )}{17 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.025, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \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}} x^{2}\,{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 x^{4} + a x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.16534, size = 29, normalized size = 0.09 \begin{align*} \frac{a^{\frac{4}{3}} x^{3}{{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3} \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}} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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