Integrand size = 24, antiderivative size = 584 \[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^2} \, dx=\frac {x \left (a-b x^2\right )^{2/3}}{6 a \left (3 a+b x^2\right )}-\frac {x}{6 a \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{4\ 3^{3/4} a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
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Time = 0.30 (sec) , antiderivative size = 584, 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 = {423, 21, 241, 310, 225, 1893} \[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^2} \, dx=\frac {\left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{3 \sqrt {2} \sqrt [4]{3} a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {\sqrt {2+\sqrt {3}} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{4\ 3^{3/4} a^{2/3} b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {x \left (a-b x^2\right )^{2/3}}{6 a \left (3 a+b x^2\right )}-\frac {x}{6 a \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )} \]
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Rule 21
Rule 225
Rule 241
Rule 310
Rule 423
Rule 1893
Rubi steps \begin{align*} \text {integral}& = \frac {x \left (a-b x^2\right )^{2/3}}{6 a \left (3 a+b x^2\right )}-\frac {\int \frac {-a-\frac {b x^2}{3}}{\sqrt [3]{a-b x^2} \left (3 a+b x^2\right )} \, dx}{6 a} \\ & = \frac {x \left (a-b x^2\right )^{2/3}}{6 a \left (3 a+b x^2\right )}+\frac {\int \frac {1}{\sqrt [3]{a-b x^2}} \, dx}{18 a} \\ & = \frac {x \left (a-b x^2\right )^{2/3}}{6 a \left (3 a+b x^2\right )}-\frac {\sqrt {-b x^2} \text {Subst}\left (\int \frac {x}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{12 a b x} \\ & = \frac {x \left (a-b x^2\right )^{2/3}}{6 a \left (3 a+b x^2\right )}+\frac {\sqrt {-b x^2} \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 )}{12 a b x}-\frac {\left (\left (1+\sqrt {3}\right ) \sqrt {-b x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a+x^3}} \, dx,x,\sqrt [3]{a-b x^2}\right )}{12 a^{2/3} b x} \\ & = \frac {x \left (a-b x^2\right )^{2/3}}{6 a \left (3 a+b x^2\right )}-\frac {x}{6 a \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {\sqrt {2+\sqrt {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 )}{4\ 3^{3/4} a^{2/3} 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 {\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 )}{3 \sqrt {2} \sqrt [4]{3} a^{2/3} 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 = 10.05 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.15 \[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^2} \, dx=\frac {x \left (a-b x^2\right )^{2/3}}{6 a \left (3 a+b x^2\right )}+\frac {x \sqrt [3]{\frac {a-b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {b x^2}{a}\right )}{18 a \sqrt [3]{a-b x^2}} \]
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\[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {2}{3}}}{\left (b \,x^{2}+3 a \right )^{2}}d x\]
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\[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^2} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {2}{3}}}{{\left (b x^{2} + 3 \, a\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^2} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {2}{3}}}{\left (3 a + b x^{2}\right )^{2}}\, dx \]
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\[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^2} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {2}{3}}}{{\left (b x^{2} + 3 \, a\right )}^{2}} \,d x } \]
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\[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^2} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {2}{3}}}{{\left (b x^{2} + 3 \, a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^2} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{2/3}}{{\left (b\,x^2+3\,a\right )}^2} \,d x \]
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