Integrand size = 24, antiderivative size = 592 \[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\frac {45}{7} x \left (a-b x^2\right )^{2/3}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}+\frac {324 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {108 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
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Time = 0.25 (sec) , antiderivative size = 592, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {424, 396, 241, 310, 225, 1893} \[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=-\frac {108 \sqrt {2} 3^{3/4} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{7 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {324 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}+\frac {45}{7} x \left (a-b x^2\right )^{2/3} \]
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Rule 225
Rule 241
Rule 310
Rule 396
Rule 424
Rule 1893
Rubi steps \begin{align*} \text {integral}& = \frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-\frac {3 \int \frac {6 a^2 b+10 a b^2 x^2}{\sqrt [3]{a-b x^2}} \, dx}{2 a b} \\ & = \frac {45}{7} x \left (a-b x^2\right )^{2/3}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-\frac {1}{7} (108 a) \int \frac {1}{\sqrt [3]{a-b x^2}} \, dx \\ & = \frac {45}{7} x \left (a-b x^2\right )^{2/3}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}+\frac {\left (162 a \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{7 b x} \\ & = \frac {45}{7} x \left (a-b x^2\right )^{2/3}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}-\frac {\left (162 a \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 )}{7 b x}+\frac {\left (162 \left (1+\sqrt {3}\right ) a^{4/3} \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{7 b x} \\ & = \frac {45}{7} x \left (a-b x^2\right )^{2/3}+\frac {6 x \left (3 a+b x^2\right )}{\sqrt [3]{a-b x^2}}+\frac {324 a x}{7 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {162 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{4/3} \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 )}{7 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 {108 \sqrt {2} 3^{3/4} a^{4/3} \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 )}{7 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 = 14.23 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.28 \[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\frac {x \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {Gamma}\left (\frac {1}{3}\right ) \left (63 a \left (45 a^2+10 a b x^2+b^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {4}{3},\frac {7}{2},\frac {b x^2}{a}\right )+32 b x^2 \left (18 a^2+9 a b x^2+b^2 x^4\right ) \operatorname {Hypergeometric2F1}\left (\frac {3}{2},\frac {7}{3},\frac {9}{2},\frac {b x^2}{a}\right )+16 b \left (3 a x+b x^3\right )^2 \, _3F_2\left (\frac {3}{2},2,\frac {7}{3};1,\frac {9}{2};\frac {b x^2}{a}\right )\right )}{945 a^2 \sqrt [3]{a-b x^2} \operatorname {Gamma}\left (\frac {4}{3}\right )} \]
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\[\int \frac {\left (b \,x^{2}+3 a \right )^{2}}{\left (-b \,x^{2}+a \right )^{\frac {4}{3}}}d x\]
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\[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\int \frac {\left (3 a + b x^{2}\right )^{2}}{\left (a - b x^{2}\right )^{\frac {4}{3}}}\, dx \]
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\[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \]
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\[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{2}}{{\left (-b x^{2} + a\right )}^{\frac {4}{3}}} \,d x } \]
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Timed out. \[ \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{4/3}} \, dx=\int \frac {{\left (b\,x^2+3\,a\right )}^2}{{\left (a-b\,x^2\right )}^{4/3}} \,d x \]
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