Integrand size = 24, antiderivative size = 617 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\frac {7776 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac {252}{247} a x \left (a-b x^2\right )^{5/3}-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {31104 a^3 x}{1729 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {15552 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{1729 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {10368 \sqrt {2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{1729 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
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Time = 0.34 (sec) , antiderivative size = 617, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {427, 396, 201, 241, 310, 225, 1893} \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\frac {10368 \sqrt {2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{1729 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {15552 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{1729 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {31104 a^3 x}{1729 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {7776 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac {252}{247} a x \left (a-b x^2\right )^{5/3}-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right ) \]
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Rule 201
Rule 225
Rule 241
Rule 310
Rule 396
Rule 427
Rule 1893
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {3 \int \left (a-b x^2\right )^{2/3} \left (-60 a^2 b-28 a b^2 x^2\right ) \, dx}{19 b} \\ & = -\frac {252}{247} a x \left (a-b x^2\right )^{5/3}-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )+\frac {1}{247} \left (2592 a^2\right ) \int \left (a-b x^2\right )^{2/3} \, dx \\ & = \frac {7776 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac {252}{247} a x \left (a-b x^2\right )^{5/3}-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )+\frac {\left (10368 a^3\right ) \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{1729} \\ & = \frac {7776 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac {252}{247} a x \left (a-b x^2\right )^{5/3}-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {\left (15552 a^3 \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{1729 b x} \\ & = \frac {7776 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac {252}{247} a x \left (a-b x^2\right )^{5/3}-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )+\frac {\left (15552 a^3 \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 )}{1729 b x}-\frac {\left (15552 \left (1+\sqrt {3}\right ) a^{10/3} \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{1729 b x} \\ & = \frac {7776 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac {252}{247} a x \left (a-b x^2\right )^{5/3}-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {31104 a^3 x}{1729 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {15552 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{10/3} \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 )}{1729 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 {10368 \sqrt {2} 3^{3/4} a^{10/3} \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 )}{1729 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 = 8.21 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.29 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\frac {x \left (a-b x^2\right )^{2/3} \left (21 a \left (45 a^2+10 a b x^2+b^2 x^4\right ) \operatorname {Gamma}\left (-\frac {2}{3}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {7}{2},\frac {b x^2}{a}\right )+8 b x^2 \left (18 a^2+9 a b x^2+b^2 x^4\right ) \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{2},\frac {9}{2},\frac {b x^2}{a}\right )+4 b \left (3 a x+b x^3\right )^2 \operatorname {Gamma}\left (\frac {1}{3}\right ) \, _3F_2\left (\frac {1}{3},\frac {3}{2},2;1,\frac {9}{2};\frac {b x^2}{a}\right )\right )}{105 a \left (1-\frac {b x^2}{a}\right )^{2/3} \operatorname {Gamma}\left (-\frac {2}{3}\right )} \]
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\[\int \left (-b \,x^{2}+a \right )^{\frac {2}{3}} \left (b \,x^{2}+3 a \right )^{2}d x\]
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\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{2} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \]
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Time = 1.44 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.16 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=9 a^{\frac {8}{3}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + 2 a^{\frac {5}{3}} b x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + \frac {a^{\frac {2}{3}} b^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} \]
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\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{2} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \]
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\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{2} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \]
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Timed out. \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\int {\left (a-b\,x^2\right )}^{2/3}\,{\left (b\,x^2+3\,a\right )}^2 \,d x \]
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