Integrand size = 52, antiderivative size = 304 \[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=\frac {5}{16} (1-x)^{3/4} \sqrt [4]{1+x}-\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {7 \left (1-x^2\right )^{5/4}}{24 \sqrt {1-x}}+\frac {x \left (1-x^2\right )^{5/4}}{6 \sqrt {1-x}}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}} \]
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Time = 0.64 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.05, number of steps used = 33, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2128, 809, 689, 52, 65, 246, 217, 1179, 642, 1176, 631, 210, 1647, 807, 338, 303} \[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{8 \sqrt {2}}+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt {2}}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}+\frac {1}{6} \sqrt {x+1} \left (1-x^2\right )^{5/4}+\frac {1}{6} (1-x)^{7/4} (x+1)^{5/4}+\frac {1}{24} (1-x)^{5/4} (x+1)^{3/4}-\frac {1}{16} \sqrt [4]{1-x} (x+1)^{3/4}+\frac {5}{24} (1-x)^{7/4} \sqrt [4]{x+1}-\frac {5}{48} (1-x)^{3/4} \sqrt [4]{x+1}+\frac {\log \left (\frac {\sqrt {1-x}}{\sqrt {x+1}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt {2}}-\frac {\log \left (\frac {\sqrt {1-x}}{\sqrt {x+1}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt {2}} \]
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Rule 52
Rule 65
Rule 210
Rule 217
Rule 246
Rule 303
Rule 338
Rule 631
Rule 642
Rule 689
Rule 807
Rule 809
Rule 1176
Rule 1179
Rule 1647
Rule 2128
Rubi steps \begin{align*} \text {integral}& = -\left (\frac {1}{2} \int x \sqrt {1+x} \sqrt [4]{1-x^2} \, dx\right )-\frac {1}{2} \int \frac {x (1+x) \sqrt [4]{1-x^2}}{\sqrt {1-x}} \, dx \\ & = \frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}-\frac {1}{12} \int \sqrt {1+x} \sqrt [4]{1-x^2} \, dx-\frac {1}{2} \int \frac {x \left (1-x^2\right )^{5/4}}{(1-x)^{3/2}} \, dx \\ & = \frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac {1}{12} \int \sqrt [4]{1-x} (1+x)^{3/4} \, dx-\frac {1}{2} \int \frac {\left (1-x^2\right )^{5/4}}{\sqrt {1-x}} \, dx \\ & = \frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac {1}{16} \int \frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}} \, dx-\frac {1}{2} \int (1-x)^{3/4} (1+x)^{5/4} \, dx \\ & = -\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac {1}{32} \int \frac {1}{(1-x)^{3/4} \sqrt [4]{1+x}} \, dx-\frac {5}{12} \int (1-x)^{3/4} \sqrt [4]{1+x} \, dx \\ & = \frac {5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac {5}{48} \int \frac {(1-x)^{3/4}}{(1+x)^{3/4}} \, dx+\frac {1}{8} \text {Subst}\left (\int \frac {1}{\sqrt [4]{2-x^4}} \, dx,x,\sqrt [4]{1-x}\right ) \\ & = -\frac {5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac {5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}+\frac {1}{8} \text {Subst}\left (\int \frac {1}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\frac {5}{32} \int \frac {1}{\sqrt [4]{1-x} (1+x)^{3/4}} \, dx \\ & = -\frac {5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac {5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}+\frac {1}{16} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac {1}{16} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac {5}{8} \text {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-x}\right ) \\ & = -\frac {5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac {5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}+\frac {1}{32} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac {1}{32} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac {5}{8} \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt {2}} \\ & = -\frac {5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac {5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt {2}}-\frac {5}{16} \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac {5}{16} \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}} \\ & = -\frac {5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac {5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt {2}}+\frac {5}{32} \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac {5}{32} \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )+\frac {5 \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt {2}}+\frac {5 \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{32 \sqrt {2}} \\ & = -\frac {5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac {5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}+\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}-\frac {5 \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}} \\ & = -\frac {5}{48} (1-x)^{3/4} \sqrt [4]{1+x}+\frac {5}{24} (1-x)^{7/4} \sqrt [4]{1+x}-\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {1}{6} (1-x)^{7/4} (1+x)^{5/4}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}+\frac {\left (1-x^2\right )^{9/4}}{3 (1-x)^{3/2}}-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}+\frac {3 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}} \\ \end{align*}
Time = 11.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.52 \[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=-\frac {1}{48} \sqrt {1+x} \sqrt [4]{1-x^2} \left (-7+2 x+8 x^2-\frac {\sqrt {1-x^2} \left (29+22 x+8 x^2\right )}{1+x}\right )+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt {1+x} \sqrt [4]{1-x^2}}{1+x-\sqrt {1-x^2}}\right )-2 \text {arctanh}\left (\frac {1+x+\sqrt {1-x^2}}{\sqrt {2} \sqrt {1+x} \sqrt [4]{1-x^2}}\right )}{8 \sqrt {2}} \]
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\[\int \frac {x^{2} \left (-x^{2}+1\right )^{\frac {1}{4}} \sqrt {1+x}}{\sqrt {1-x}\, \left (\sqrt {1-x}-\sqrt {1+x}\right )}d x\]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.21 \[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=-\frac {1}{48} \, {\left (8 \, x^{2} + 2 \, x - 7\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} + \frac {1}{48} \, {\left (8 \, x^{2} + 22 \, x + 29\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} + \left (\frac {1}{64} i + \frac {1}{64}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i + 1\right ) \, x + i + 1\right )} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1}}{x + 1}\right ) - \left (\frac {1}{64} i - \frac {1}{64}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i - 1\right ) \, x - i + 1\right )} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1}}{x + 1}\right ) + \left (\frac {1}{64} i - \frac {1}{64}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i - 1\right ) \, x + i - 1\right )} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1}}{x + 1}\right ) - \left (\frac {1}{64} i + \frac {1}{64}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i + 1\right ) \, x - i - 1\right )} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1}}{x + 1}\right ) + \left (\frac {5}{64} i + \frac {5}{64}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i + 1\right ) \, x - i - 1\right )} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1}}{x - 1}\right ) - \left (\frac {5}{64} i - \frac {5}{64}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i - 1\right ) \, x + i - 1\right )} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1}}{x - 1}\right ) + \left (\frac {5}{64} i - \frac {5}{64}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (\left (i - 1\right ) \, x - i + 1\right )} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1}}{x - 1}\right ) - \left (\frac {5}{64} i + \frac {5}{64}\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2} {\left (-\left (i + 1\right ) \, x + i + 1\right )} + 2 \, {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1}}{x - 1}\right ) \]
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\[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=\int \frac {x^{2} \sqrt [4]{- \left (x - 1\right ) \left (x + 1\right )} \sqrt {x + 1}}{\sqrt {1 - x} \left (\sqrt {1 - x} - \sqrt {x + 1}\right )}\, dx \]
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\[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=\int { -\frac {{\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} x^{2}}{\sqrt {-x + 1} {\left (\sqrt {x + 1} - \sqrt {-x + 1}\right )}} \,d x } \]
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\[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=\int { -\frac {{\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} x^{2}}{\sqrt {-x + 1} {\left (\sqrt {x + 1} - \sqrt {-x + 1}\right )}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx=-\int \frac {x^2\,{\left (1-x^2\right )}^{1/4}\,\sqrt {x+1}}{\left (\sqrt {x+1}-\sqrt {1-x}\right )\,\sqrt {1-x}} \,d x \]
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