Integrand size = 24, antiderivative size = 648 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\frac {18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac {23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2-\frac {72576 a^4 x}{1235 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {36288 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{1235 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {24192 \sqrt {2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{1235 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
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Time = 0.43 (sec) , antiderivative size = 648, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {427, 542, 396, 201, 241, 310, 225, 1893} \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\frac {24192 \sqrt {2} 3^{3/4} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{1235 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {36288 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{13/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{1235 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {72576 a^4 x}{1235 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac {23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2 \]
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Rule 201
Rule 225
Rule 241
Rule 310
Rule 396
Rule 427
Rule 542
Rule 1893
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2-\frac {3 \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right ) \left (-78 a^2 b-42 a b^2 x^2\right ) \, dx}{25 b} \\ & = -\frac {378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2+\frac {9 \int \left (a-b x^2\right )^{2/3} \left (1608 a^3 b^2+872 a^2 b^3 x^2\right ) \, dx}{475 b^2} \\ & = -\frac {23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2+\frac {\left (42336 a^3\right ) \int \left (a-b x^2\right )^{2/3} \, dx}{1235} \\ & = \frac {18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac {23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2+\frac {\left (24192 a^4\right ) \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{1235} \\ & = \frac {18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac {23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2-\frac {\left (36288 a^4 \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{1235 b x} \\ & = \frac {18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac {23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2+\frac {\left (36288 a^4 \sqrt {-b x^2}\right ) \text {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 )}{1235 b x}-\frac {\left (36288 \left (1+\sqrt {3}\right ) a^{13/3} \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{1235 b x} \\ & = \frac {18144 a^3 x \left (a-b x^2\right )^{2/3}}{1235}-\frac {23544 a^2 x \left (a-b x^2\right )^{5/3}}{6175}-\frac {378}{475} a x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {3}{25} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^2-\frac {72576 a^4 x}{1235 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {36288 \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 )}{1235 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 {24192 \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 )}{1235 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*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 11.11 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.15 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=-\frac {3 \left (-15255 a^4 x+3390 a^3 b x^3+8992 a^2 b^2 x^5+2626 a b^3 x^7+247 b^4 x^9-40320 a^4 x \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {b x^2}{a}\right )\right )}{6175 \sqrt [3]{a-b x^2}} \]
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\[\int \left (-b \,x^{2}+a \right )^{\frac {2}{3}} \left (b \,x^{2}+3 a \right )^{3}d x\]
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\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{3} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \]
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Time = 1.86 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.21 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=27 a^{\frac {11}{3}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + 9 a^{\frac {8}{3}} b x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + \frac {9 a^{\frac {5}{3}} b^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} + \frac {a^{\frac {2}{3}} b^{3} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{7} \]
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\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{3} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \]
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\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{3} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \]
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Timed out. \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^3 \, dx=\int {\left (a-b\,x^2\right )}^{2/3}\,{\left (b\,x^2+3\,a\right )}^3 \,d x \]
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