Optimal. Leaf size=740 \[ -\frac{\sqrt{2} 3^{3/4} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt{3}-7\right )}{b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{2 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{3 x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{\sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}-\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}} \]
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Rubi [A] time = 0.320949, antiderivative size = 740, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {396, 235, 304, 219, 1879, 393} \[ -\frac{\sqrt{2} 3^{3/4} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{2 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{3 x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{\sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}-\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}} \]
Antiderivative was successfully verified.
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Rule 396
Rule 235
Rule 304
Rule 219
Rule 1879
Rule 393
Rubi steps
\begin{align*} \int \frac{\left (a-b x^2\right )^{2/3}}{3 a+b x^2} \, dx &=(4 a) \int \frac{1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx-\int \frac{1}{\sqrt [3]{a-b x^2}} \, dx\\ &=\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}-\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{\sqrt{b}}+\frac{\left (3 \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{2 b x}\\ &=\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}-\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{\sqrt{b}}-\frac{\left (3 \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{2 b x}+\frac{\left (3 \sqrt{\frac{1}{2} \left (2+\sqrt{3}\right )} \sqrt [3]{a} \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{b x}\\ &=\frac{3 x}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}-\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}}+\frac{\sqrt [3]{2} \sqrt [6]{a} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{\sqrt{b}}+\frac{3 \sqrt [4]{3} \sqrt{2+\sqrt{3}} \sqrt [3]{a} \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}} E\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 )}{2 b 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{\sqrt{2} 3^{3/4} \sqrt [3]{a} \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 )}{b 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 = 0.150068, size = 162, normalized size = 0.22 \[ \frac{9 a x \left (a-b x^2\right )^{2/3} F_1\left (\frac{1}{2};-\frac{2}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{\left (3 a+b x^2\right ) \left (9 a F_1\left (\frac{1}{2};-\frac{2}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-2 b x^2 \left (F_1\left (\frac{3}{2};-\frac{2}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )+2 F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.047, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{2}+3\,a} \left ( -b{x}^{2}+a \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{2}{3}}}{b x^{2} + 3 \, a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a - b x^{2}\right )^{\frac{2}{3}}}{3 a + b x^{2}}\, dx \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{2}{3}}}{b x^{2} + 3 \, a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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