Optimal. Leaf size=659 \[ \frac{1264896 \sqrt{2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt{3}-7\right )}{8645 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{3794688 a^4 x}{8645 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac{36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac{1897344 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{8645 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac{3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \]
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Rubi [A] time = 0.52655, antiderivative size = 659, 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, 528, 388, 235, 304, 219, 1879} \[ -\frac{3794688 a^4 x}{8645 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac{36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}+\frac{1264896 \sqrt{2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{8645 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{1897344 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{8645 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac{3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \]
Antiderivative was successfully verified.
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Rule 416
Rule 528
Rule 388
Rule 235
Rule 304
Rule 219
Rule 1879
Rubi steps
\begin{align*} \int \frac{\left (3 a+b x^2\right )^4}{\sqrt [3]{a-b x^2}} \, dx &=-\frac{3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac{3 \int \frac{\left (3 a+b x^2\right )^2 \left (-78 a^2 b-50 a b^2 x^2\right )}{\sqrt [3]{a-b x^2}} \, dx}{25 b}\\ &=-\frac{18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac{3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3+\frac{9 \int \frac{\left (3 a+b x^2\right ) \left (1632 a^3 b^2+1344 a^2 b^3 x^2\right )}{\sqrt [3]{a-b x^2}} \, dx}{475 b^2}\\ &=-\frac{36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac{18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac{3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac{27 \int \frac{-25248 a^4 b^3-19168 a^3 b^4 x^2}{\sqrt [3]{a-b x^2}} \, dx}{6175 b^3}\\ &=-\frac{1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac{36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac{18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac{3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3+\frac{\left (1264896 a^4\right ) \int \frac{1}{\sqrt [3]{a-b x^2}} \, dx}{8645}\\ &=-\frac{1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac{36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac{18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac{3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac{\left (1897344 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{1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac{36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac{18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac{3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3+\frac{\left (1897344 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 (1897344 \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{1552608 a^3 x \left (a-b x^2\right )^{2/3}}{43225}-\frac{36288 a^2 x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )}{6175}-\frac{18}{19} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2-\frac{3}{25} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3-\frac{3794688 a^4 x}{8645 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{1897344 \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{1264896 \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 = 5.05644, size = 98, normalized size = 0.15 \[ \frac{3 x \left (184044 a^2 b^2 x^4+2108160 a^4 \sqrt [3]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )+727830 a^3 b x^2-941085 a^4+27482 a b^3 x^6+1729 b^4 x^8\right )}{43225 \sqrt [3]{a-b x^2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{2}+3\,a \right ) ^{4}{\frac{1}{\sqrt [3]{-b{x}^{2}+a}}}}\, 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} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac{1}{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 (-\frac{{\left (b^{4} x^{8} + 12 \, a b^{3} x^{6} + 54 \, a^{2} b^{2} x^{4} + 108 \, a^{3} b x^{2} + 81 \, a^{4}\right )}{\left (-b x^{2} + a\right )}^{\frac{2}{3}}}{b x^{2} - a}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.45997, size = 165, normalized size = 0.25 \begin{align*} 81 a^{\frac{11}{3}} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} + 36 a^{\frac{8}{3}} b x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} + \frac{54 a^{\frac{5}{3}} b^{2} x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5} + \frac{12 a^{\frac{2}{3}} b^{3} x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{7} + \frac{b^{4} x^{9}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{9}{2} \\ \frac{11}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{9 \sqrt [3]{a}} \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} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac{1}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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