Optimal. Leaf size=637 \[ \frac{38016 \sqrt{2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt{3}-7\right )}{8645 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{114048 a^4 x}{8645 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac{14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{57024 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{8645 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{306}{475} a x \left (a-b x^2\right )^{8/3}-\frac{3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right ) \]
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Rubi [A] time = 0.483626, antiderivative size = 637, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {416, 388, 195, 235, 304, 219, 1879} \[ -\frac{114048 a^4 x}{8645 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac{14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}+\frac{38016 \sqrt{2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{8645 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{57024 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{8645 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{306}{475} a x \left (a-b x^2\right )^{8/3}-\frac{3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right ) \]
Antiderivative was successfully verified.
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Rule 416
Rule 388
Rule 195
Rule 235
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2 \, dx &=-\frac{3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3 \int \left (a-b x^2\right )^{5/3} \left (-78 a^2 b-34 a b^2 x^2\right ) \, dx}{25 b}\\ &=-\frac{306}{475} a x \left (a-b x^2\right )^{8/3}-\frac{3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )+\frac{1}{475} \left (4752 a^2\right ) \int \left (a-b x^2\right )^{5/3} \, dx\\ &=\frac{14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{306}{475} a x \left (a-b x^2\right )^{8/3}-\frac{3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )+\frac{\left (9504 a^3\right ) \int \left (a-b x^2\right )^{2/3} \, dx}{1235}\\ &=\frac{28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac{14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{306}{475} a x \left (a-b x^2\right )^{8/3}-\frac{3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )+\frac{\left (38016 a^4\right ) \int \frac{1}{\sqrt [3]{a-b x^2}} \, dx}{8645}\\ &=\frac{28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac{14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{306}{475} a x \left (a-b x^2\right )^{8/3}-\frac{3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{\left (57024 a^4 \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{8645 b x}\\ &=\frac{28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac{14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{306}{475} a x \left (a-b x^2\right )^{8/3}-\frac{3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )+\frac{\left (57024 a^4 \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 )}{8645 b x}-\frac{\left (57024 \sqrt{2 \left (2+\sqrt{3}\right )} a^{13/3} \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{8645 b x}\\ &=\frac{28512 a^3 x \left (a-b x^2\right )^{2/3}}{8645}+\frac{14256 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac{306}{475} a x \left (a-b x^2\right )^{8/3}-\frac{3}{25} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{114048 a^4 x}{8645 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{57024 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \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 )}{8645 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{38016 \sqrt{2} 3^{3/4} a^{13/3} \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 )}{8645 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 = 2.77532, size = 173, normalized size = 0.27 \[ \frac{x \left (a-b x^2\right )^{2/3} \left (4 b \text{Gamma}\left (-\frac{2}{3}\right ) \left (3 a x+b x^3\right )^2 \text{HypergeometricPFQ}\left (\left \{-\frac{2}{3},\frac{3}{2},2\right \},\left \{1,\frac{9}{2}\right \},\frac{b x^2}{a}\right )+8 b x^2 \text{Gamma}\left (-\frac{2}{3}\right ) \left (18 a^2+9 a b x^2+b^2 x^4\right ) \, _2F_1\left (-\frac{2}{3},\frac{3}{2};\frac{9}{2};\frac{b x^2}{a}\right )+21 a \text{Gamma}\left (-\frac{5}{3}\right ) \left (45 a^2+10 a b x^2+b^2 x^4\right ) \, _2F_1\left (-\frac{5}{3},\frac{1}{2};\frac{7}{2};\frac{b x^2}{a}\right )\right )}{105 \text{Gamma}\left (-\frac{5}{3}\right ) \left (1-\frac{b x^2}{a}\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.027, size = 0, normalized size = 0. \begin{align*} \int \left ( -b{x}^{2}+a \right ) ^{{\frac{5}{3}}} \left ( b{x}^{2}+3\,a \right ) ^{2}\, 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} + 3 \, a\right )}^{2}{\left (-b x^{2} + a\right )}^{\frac{5}{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^{3} x^{6} + 5 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - 9 \, a^{3}\right )}{\left (-b x^{2} + a\right )}^{\frac{2}{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.90736, size = 131, normalized size = 0.21 \begin{align*} 9 a^{\frac{11}{3}} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} - a^{\frac{8}{3}} b x^{3}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} - a^{\frac{5}{3}} b^{2} x^{5}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} - \frac{a^{\frac{2}{3}} b^{3} x^{7}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{7} \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} + 3 \, a\right )}^{2}{\left (-b x^{2} + a\right )}^{\frac{5}{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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