Optimal. Leaf size=668 \[ \frac{3746304 \sqrt{2} 3^{3/4} a^{16/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),4 \sqrt{3}-7\right )}{267995 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{11238912 a^5 x}{267995 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{2809728 a^4 x \left (a-b x^2\right )^{2/3}}{267995}+\frac{1404864 a^3 x \left (a-b x^2\right )^{5/3}}{191425}-\frac{33264 a^2 x \left (a-b x^2\right )^{8/3}}{14725}-\frac{5619456 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{16/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{267995 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2 \]
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Rubi [A] time = 0.570797, antiderivative size = 668, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {416, 528, 388, 195, 235, 304, 219, 1879} \[ -\frac{11238912 a^5 x}{267995 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{2809728 a^4 x \left (a-b x^2\right )^{2/3}}{267995}+\frac{1404864 a^3 x \left (a-b x^2\right )^{5/3}}{191425}-\frac{33264 a^2 x \left (a-b x^2\right )^{8/3}}{14725}+\frac{3746304 \sqrt{2} 3^{3/4} a^{16/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{267995 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{5619456 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{16/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{267995 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2 \]
Antiderivative was successfully verified.
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Rule 416
Rule 528
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 )^3 \, dx &=-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2-\frac{3 \int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right ) \left (-96 a^2 b-48 a b^2 x^2\right ) \, dx}{31 b}\\ &=-\frac{432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2+\frac{9 \int \left (a-b x^2\right )^{5/3} \left (2544 a^3 b^2+1232 a^2 b^3 x^2\right ) \, dx}{775 b^2}\\ &=-\frac{33264 a^2 x \left (a-b x^2\right )^{8/3}}{14725}-\frac{432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2+\frac{\left (468288 a^3\right ) \int \left (a-b x^2\right )^{5/3} \, dx}{14725}\\ &=\frac{1404864 a^3 x \left (a-b x^2\right )^{5/3}}{191425}-\frac{33264 a^2 x \left (a-b x^2\right )^{8/3}}{14725}-\frac{432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2+\frac{\left (936576 a^4\right ) \int \left (a-b x^2\right )^{2/3} \, dx}{38285}\\ &=\frac{2809728 a^4 x \left (a-b x^2\right )^{2/3}}{267995}+\frac{1404864 a^3 x \left (a-b x^2\right )^{5/3}}{191425}-\frac{33264 a^2 x \left (a-b x^2\right )^{8/3}}{14725}-\frac{432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2+\frac{\left (3746304 a^5\right ) \int \frac{1}{\sqrt [3]{a-b x^2}} \, dx}{267995}\\ &=\frac{2809728 a^4 x \left (a-b x^2\right )^{2/3}}{267995}+\frac{1404864 a^3 x \left (a-b x^2\right )^{5/3}}{191425}-\frac{33264 a^2 x \left (a-b x^2\right )^{8/3}}{14725}-\frac{432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2-\frac{\left (5619456 a^5 \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{267995 b x}\\ &=\frac{2809728 a^4 x \left (a-b x^2\right )^{2/3}}{267995}+\frac{1404864 a^3 x \left (a-b x^2\right )^{5/3}}{191425}-\frac{33264 a^2 x \left (a-b x^2\right )^{8/3}}{14725}-\frac{432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2+\frac{\left (5619456 a^5 \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 )}{267995 b x}-\frac{\left (5619456 \sqrt{2 \left (2+\sqrt{3}\right )} a^{16/3} \sqrt{-b x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{267995 b x}\\ &=\frac{2809728 a^4 x \left (a-b x^2\right )^{2/3}}{267995}+\frac{1404864 a^3 x \left (a-b x^2\right )^{5/3}}{191425}-\frac{33264 a^2 x \left (a-b x^2\right )^{8/3}}{14725}-\frac{432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2-\frac{11238912 a^5 x}{267995 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{5619456 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{16/3} \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 )}{267995 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{3746304 \sqrt{2} 3^{3/4} a^{16/3} \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 )}{267995 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 = 5.04916, size = 110, normalized size = 0.16 \[ \frac{3 \left (749658 a^2 b^3 x^7-1675114 a^3 b^2 x^5+6243840 a^5 x \sqrt [3]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )-5312355 a^4 b x^3+5815935 a^5 x+378651 a b^4 x^9+43225 b^5 x^{11}\right )}{1339975 \sqrt [3]{a-b x^2}} \]
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}+a \right ) ^{{\frac{5}{3}}} \left ( b{x}^{2}+3\,a \right ) ^{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} + 3 \, a\right )}^{3}{\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^{4} x^{8} + 8 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} - 27 \, a^{4}\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 = 6.24803, size = 139, normalized size = 0.21 \begin{align*} 27 a^{\frac{14}{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 )} - \frac{18 a^{\frac{8}{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 )}}{5} - \frac{8 a^{\frac{5}{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} - \frac{a^{\frac{2}{3}} b^{4} x^{9}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{9}{2} \\ \frac{11}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{9} \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 )}^{3}{\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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