Integrand size = 22, antiderivative size = 543 \[ \int \frac {x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=-\frac {x \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac {x}{8 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt {3}}-\frac {\text {arctanh}(x)}{24\ 2^{2/3}}+\frac {\text {arctanh}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{8\ 2^{2/3}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{16 x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}-\frac {\left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right ),-7+4 \sqrt {3}\right )}{4 \sqrt {2} \sqrt [4]{3} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}} \]
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Time = 0.18 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {482, 544, 241, 310, 225, 1893, 402} \[ \int \frac {x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=-\frac {\left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{4 \sqrt {2} \sqrt [4]{3} \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {\left (1-x^2\right )^{2/3}+\sqrt [3]{1-x^2}+1}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-\sqrt [3]{1-x^2}+\sqrt {3}+1}{-\sqrt [3]{1-x^2}-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{16 \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )^2}} x}+\frac {\arctan \left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt {3}}+\frac {\arctan \left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3} \sqrt {3}}+\frac {\text {arctanh}\left (\frac {x}{\sqrt [3]{2} \sqrt [3]{1-x^2}+1}\right )}{8\ 2^{2/3}}-\frac {\text {arctanh}(x)}{24\ 2^{2/3}}-\frac {\left (1-x^2\right )^{2/3} x}{8 \left (x^2+3\right )}+\frac {x}{8 \left (-\sqrt [3]{1-x^2}-\sqrt {3}+1\right )} \]
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Rule 225
Rule 241
Rule 310
Rule 402
Rule 482
Rule 544
Rule 1893
Rubi steps \begin{align*} \text {integral}& = -\frac {x \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac {1}{8} \int \frac {1-\frac {x^2}{3}}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx \\ & = -\frac {x \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}-\frac {1}{24} \int \frac {1}{\sqrt [3]{1-x^2}} \, dx+\frac {1}{4} \int \frac {1}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx \\ & = -\frac {x \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}(x)}{24\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{8\ 2^{2/3}}+\frac {\sqrt {-x^2} \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{16 x} \\ & = -\frac {x \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}(x)}{24\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{8\ 2^{2/3}}-\frac {\sqrt {-x^2} \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{16 x}+\frac {\left (\left (1+\sqrt {3}\right ) \sqrt {-x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1-x^2}\right )}{16 x} \\ & = -\frac {x \left (1-x^2\right )^{2/3}}{8 \left (3+x^2\right )}+\frac {x}{8 \left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {3}}{x}\right )}{8\ 2^{2/3} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{2} \sqrt [3]{1-x^2}\right )}{x}\right )}{8\ 2^{2/3} \sqrt {3}}-\frac {\tanh ^{-1}(x)}{24\ 2^{2/3}}+\frac {\tanh ^{-1}\left (\frac {x}{1+\sqrt [3]{2} \sqrt [3]{1-x^2}}\right )}{8\ 2^{2/3}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{16 x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}}-\frac {\left (1-\sqrt [3]{1-x^2}\right ) \sqrt {\frac {1+\sqrt [3]{1-x^2}+\left (1-x^2\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-\sqrt [3]{1-x^2}}{1-\sqrt {3}-\sqrt [3]{1-x^2}}\right )|-7+4 \sqrt {3}\right )}{4 \sqrt {2} \sqrt [4]{3} x \sqrt {-\frac {1-\sqrt [3]{1-x^2}}{\left (1-\sqrt {3}-\sqrt [3]{1-x^2}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 4.46 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.29 \[ \int \frac {x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=-\frac {1}{216} x^3 \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )+\frac {x \left (-1+x^2+\frac {9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )}{9 \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},x^2,-\frac {x^2}{3}\right )+2 x^2 \left (-\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},x^2,-\frac {x^2}{3}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},x^2,-\frac {x^2}{3}\right )\right )}\right )}{8 \sqrt [3]{1-x^2} \left (3+x^2\right )} \]
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\[\int \frac {x^{2}}{\left (-x^{2}+1\right )^{\frac {1}{3}} \left (x^{2}+3\right )^{2}}d x\]
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\[ \int \frac {x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int { \frac {x^{2}}{{\left (x^{2} + 3\right )}^{2} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int \frac {x^{2}}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )^{2}}\, dx \]
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\[ \int \frac {x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int { \frac {x^{2}}{{\left (x^{2} + 3\right )}^{2} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]
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\[ \int \frac {x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int { \frac {x^{2}}{{\left (x^{2} + 3\right )}^{2} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\sqrt [3]{1-x^2} \left (3+x^2\right )^2} \, dx=\int \frac {x^2}{{\left (1-x^2\right )}^{1/3}\,{\left (x^2+3\right )}^2} \,d x \]
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