Optimal. Leaf size=765 \[ -\frac{32 \sqrt{2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt{3}-7\right )}{7 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{\sqrt{b}}+\frac{48 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{7 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}-\frac{4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}}+\frac{96 a x}{7 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{3}{7} x \left (a-b x^2\right )^{2/3} \]
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Rubi [A] time = 0.443618, antiderivative size = 765, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {416, 530, 235, 304, 219, 1879, 393} \[ \frac{4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{\sqrt{b}}-\frac{32 \sqrt{2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{7 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{48 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{7 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac{4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}-\frac{4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}}+\frac{96 a x}{7 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{3}{7} x \left (a-b x^2\right )^{2/3} \]
Antiderivative was successfully verified.
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Rule 416
Rule 530
Rule 235
Rule 304
Rule 219
Rule 1879
Rule 393
Rubi steps
\begin{align*} \int \frac{\left (a-b x^2\right )^{5/3}}{3 a+b x^2} \, dx &=-\frac{3}{7} x \left (a-b x^2\right )^{2/3}+\frac{3 \int \frac{\frac{16 a^2 b}{3}-\frac{32}{3} a b^2 x^2}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{7 b}\\ &=-\frac{3}{7} x \left (a-b x^2\right )^{2/3}-\frac{1}{7} (32 a) \int \frac{1}{\sqrt [3]{a-b x^2}} \, dx+\left (16 a^2\right ) \int \frac{1}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx\\ &=-\frac{3}{7} x \left (a-b x^2\right )^{2/3}+\frac{4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{4 \sqrt [3]{2} a^{7/6} \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{4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}}+\frac{4 \sqrt [3]{2} a^{7/6} \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 (48 a \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{7 b x}\\ &=-\frac{3}{7} x \left (a-b x^2\right )^{2/3}+\frac{4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{4 \sqrt [3]{2} a^{7/6} \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{4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}}+\frac{4 \sqrt [3]{2} a^{7/6} \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 (48 a \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 )}{7 b x}+\frac{\left (48 \sqrt{2 \left (2+\sqrt{3}\right )} a^{4/3} \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{7 b x}\\ &=-\frac{3}{7} x \left (a-b x^2\right )^{2/3}+\frac{96 a x}{7 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{4 \sqrt [3]{2} a^{7/6} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{\sqrt{3} \sqrt{b}}+\frac{4 \sqrt [3]{2} a^{7/6} \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{4 \sqrt [3]{2} a^{7/6} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{3 \sqrt{b}}+\frac{4 \sqrt [3]{2} a^{7/6} \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{48 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{4/3} \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}} 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 )}{7 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{32 \sqrt{2} 3^{3/4} a^{4/3} \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 )}{7 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 = 0.133257, size = 231, normalized size = 0.3 \[ \frac{x \left (27 \left (\frac{48 a^3 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )}{\left (3 a+b x^2\right ) \left (2 b x^2 \left (F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )}-a+b x^2\right )-32 b x^2 \sqrt [3]{1-\frac{b x^2}{a}} F_1\left (\frac{3}{2};\frac{1}{3},1;\frac{5}{2};\frac{b x^2}{a},-\frac{b x^2}{3 a}\right )\right )}{63 \sqrt [3]{a-b x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{2}+3\,a} \left ( -b{x}^{2}+a \right ) ^{{\frac{5}{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{5}{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{5}{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{5}{3}}}{b x^{2} + 3 \, a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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