Optimal. Leaf size=304 \[ \frac {5}{16} (1-x)^{3/4} \sqrt [4]{1+x}-\frac {1}{16} \sqrt [4]{1-x} (1+x)^{3/4}+\frac {1}{24} (1-x)^{5/4} (1+x)^{3/4}+\frac {7 \left (1-x^2\right )^{5/4}}{24 \sqrt {1-x}}+\frac {x \left (1-x^2\right )^{5/4}}{6 \sqrt {1-x}}+\frac {1}{6} \sqrt {1+x} \left (1-x^2\right )^{5/4}-\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}+\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}-\frac {\log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}} \]
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Rubi [A]
time = 0.57, antiderivative size = 319, normalized size of antiderivative = 1.05, number of
steps used = 33, number of rules used = 16, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used
= {2128, 809, 689, 52, 65, 246, 217, 1179, 642, 1176, 631, 210, 1647, 807, 338, 303}
\begin {gather*} -\frac {3 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{8 \sqrt {2}}+\frac {3 \text {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}} \end {gather*}
Antiderivative was successfully verified.
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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*} \int \frac {x^2 \sqrt {1+x} \sqrt [4]{1-x^2}}{\sqrt {1-x} \left (\sqrt {1-x}-\sqrt {1+x}\right )} \, dx &=-\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 {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}+\frac {\tan ^{-1}\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 {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}+\frac {\tan ^{-1}\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 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}+\frac {3 \tan ^{-1}\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*}
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Mathematica [A]
time = 15.94, size = 195, normalized size = 0.64 \begin {gather*} -\frac {\sqrt [4]{1+x} \left ((-1+x) \sqrt [4]{1+x} \left (7-8 x^3+29 \sqrt {1-x^2}+2 x^2 \left (-5+4 \sqrt {1-x^2}\right )+x \left (5+22 \sqrt {1-x^2}\right )\right )+3 \sqrt [4]{-1+x} \left (\sqrt {-1+x} \sqrt {1+x}-5 \sqrt {1-x^2}\right ) \tan ^{-1}\left (\sqrt [4]{\frac {-1+x}{1+x}}\right )+3 \sqrt [4]{-1+x} \left (\sqrt {-1+x} \sqrt {1+x}+5 \sqrt {1-x^2}\right ) \tanh ^{-1}\left (\sqrt [4]{\frac {-1+x}{1+x}}\right )\right )}{48 \left (1-x^2\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{2} \left (-x^{2}+1\right )^{\frac {1}{4}} \sqrt {1+x}}{\sqrt {1-x}\, \left (\sqrt {1-x}-\sqrt {1+x}\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 577 vs.
\(2 (219) = 438\).
time = 0.46, size = 577, normalized size = 1.90 \begin {gather*} -\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} - \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x + 1\right )} \sqrt {\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} + x + \sqrt {-x^{2} + 1} + 1}{x + 1}} - \sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} - x - 1}{x + 1}\right ) - \frac {1}{16} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x + 1\right )} \sqrt {-\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} - x - \sqrt {-x^{2} + 1} - 1}{x + 1}} - \sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} + x + 1}{x + 1}\right ) - \frac {5}{16} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 1\right )} \sqrt {\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} + x - \sqrt {-x^{2} + 1} - 1}{x - 1}} - \sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} - x + 1}{x - 1}\right ) - \frac {5}{16} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 1\right )} \sqrt {-\frac {\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} - x + \sqrt {-x^{2} + 1} + 1}{x - 1}} - \sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} + x - 1}{x - 1}\right ) + \frac {1}{64} \, \sqrt {2} \log \left (\frac {4 \, {\left (\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} + x + \sqrt {-x^{2} + 1} + 1\right )}}{x + 1}\right ) - \frac {1}{64} \, \sqrt {2} \log \left (-\frac {4 \, {\left (\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {x + 1} - x - \sqrt {-x^{2} + 1} - 1\right )}}{x + 1}\right ) + \frac {5}{64} \, \sqrt {2} \log \left (\frac {4 \, {\left (\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} + x - \sqrt {-x^{2} + 1} - 1\right )}}{x - 1}\right ) - \frac {5}{64} \, \sqrt {2} \log \left (-\frac {4 \, {\left (\sqrt {2} {\left (-x^{2} + 1\right )}^{\frac {1}{4}} \sqrt {-x + 1} - x + \sqrt {-x^{2} + 1} + 1\right )}}{x - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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