Optimal. Leaf size=234 \[ -\frac {1}{3} (1-x)^{3/4} x (x+1)^{5/4}-\frac {1}{12} (1-x)^{3/4} (x+1)^{5/4}-\frac {3}{8} (1-x)^{3/4} \sqrt [4]{x+1}-\frac {3 \log \left (\frac {\sqrt {1-x}}{\sqrt {x+1}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{16 \sqrt {2}}+\frac {3 \log \left (\frac {\sqrt {1-x}}{\sqrt {x+1}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{16 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{8 \sqrt {2}}-\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt {2}} \]
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Rubi [A] time = 0.16, antiderivative size = 234, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {90, 80, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \begin {gather*} -\frac {1}{3} (1-x)^{3/4} x (x+1)^{5/4}-\frac {1}{12} (1-x)^{3/4} (x+1)^{5/4}-\frac {3}{8} (1-x)^{3/4} \sqrt [4]{x+1}-\frac {3 \log \left (\frac {\sqrt {1-x}}{\sqrt {x+1}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{16 \sqrt {2}}+\frac {3 \log \left (\frac {\sqrt {1-x}}{\sqrt {x+1}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{16 \sqrt {2}}+\frac {3 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{8 \sqrt {2}}-\frac {3 \tan ^{-1}\left (\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 50
Rule 63
Rule 80
Rule 90
Rule 204
Rule 297
Rule 331
Rule 617
Rule 628
Rule 1162
Rule 1165
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx &=-\frac {1}{3} (1-x)^{3/4} x (1+x)^{5/4}-\frac {1}{3} \int \frac {\left (-1-\frac {x}{2}\right ) \sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx\\ &=-\frac {1}{12} (1-x)^{3/4} (1+x)^{5/4}-\frac {1}{3} (1-x)^{3/4} x (1+x)^{5/4}+\frac {3}{8} \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}} \, dx\\ &=-\frac {3}{8} (1-x)^{3/4} \sqrt [4]{1+x}-\frac {1}{12} (1-x)^{3/4} (1+x)^{5/4}-\frac {1}{3} (1-x)^{3/4} x (1+x)^{5/4}+\frac {3}{16} \int \frac {1}{\sqrt [4]{1-x} (1+x)^{3/4}} \, dx\\ &=-\frac {3}{8} (1-x)^{3/4} \sqrt [4]{1+x}-\frac {1}{12} (1-x)^{3/4} (1+x)^{5/4}-\frac {1}{3} (1-x)^{3/4} x (1+x)^{5/4}-\frac {3}{4} \operatorname {Subst}\left (\int \frac {x^2}{\left (2-x^4\right )^{3/4}} \, dx,x,\sqrt [4]{1-x}\right )\\ &=-\frac {3}{8} (1-x)^{3/4} \sqrt [4]{1+x}-\frac {1}{12} (1-x)^{3/4} (1+x)^{5/4}-\frac {1}{3} (1-x)^{3/4} x (1+x)^{5/4}-\frac {3}{4} \operatorname {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )\\ &=-\frac {3}{8} (1-x)^{3/4} \sqrt [4]{1+x}-\frac {1}{12} (1-x)^{3/4} (1+x)^{5/4}-\frac {1}{3} (1-x)^{3/4} x (1+x)^{5/4}+\frac {3}{8} \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )\\ &=-\frac {3}{8} (1-x)^{3/4} \sqrt [4]{1+x}-\frac {1}{12} (1-x)^{3/4} (1+x)^{5/4}-\frac {1}{3} (1-x)^{3/4} x (1+x)^{5/4}-\frac {3}{16} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\frac {3}{16} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\frac {\sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )-\frac {3 \operatorname {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 )}{16 \sqrt {2}}-\frac {3 \operatorname {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 )}{16 \sqrt {2}}\\ &=-\frac {3}{8} (1-x)^{3/4} \sqrt [4]{1+x}-\frac {1}{12} (1-x)^{3/4} (1+x)^{5/4}-\frac {1}{3} (1-x)^{3/4} x (1+x)^{5/4}-\frac {3 \log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}+\frac {3 \log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}+\frac {3 \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{8 \sqrt {2}}\\ &=-\frac {3}{8} (1-x)^{3/4} \sqrt [4]{1+x}-\frac {1}{12} (1-x)^{3/4} (1+x)^{5/4}-\frac {1}{3} (1-x)^{3/4} x (1+x)^{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 {3 \log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}-\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}+\frac {3 \log \left (1+\frac {\sqrt {1-x}}{\sqrt {1+x}}+\frac {\sqrt {2} \sqrt [4]{1-x}}{\sqrt [4]{1+x}}\right )}{16 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 58, normalized size = 0.25 \begin {gather*} -\frac {1}{12} (1-x)^{3/4} \left (6 \sqrt [4]{2} \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};\frac {1-x}{2}\right )+\sqrt [4]{x+1} \left (4 x^2+5 x+1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.66, size = 146, normalized size = 0.62 \begin {gather*} \frac {1}{24} (1-x)^{3/4} \left (-8 (x+1)^{9/4}+6 (x+1)^{5/4}-9 \sqrt [4]{x+1}\right )+\frac {3 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-x} \sqrt [4]{x+1}}{\sqrt {1-x}-\sqrt {x+1}}\right )}{8 \sqrt {2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{1-x} \sqrt [4]{x+1}}{\sqrt {1-x}+\sqrt {x+1}}\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.26, size = 282, normalized size = 1.21 \begin {gather*} -\frac {1}{24} \, {\left (8 \, x^{2} + 10 \, x + 11\right )} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 1\right )} \sqrt {\frac {\sqrt {2} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + x - \sqrt {x + 1} \sqrt {-x + 1} - 1}{x - 1}} - \sqrt {2} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - x + 1}{x - 1}\right ) + \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 1\right )} \sqrt {-\frac {\sqrt {2} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - x + \sqrt {x + 1} \sqrt {-x + 1} + 1}{x - 1}} - \sqrt {2} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + x - 1}{x - 1}\right ) - \frac {3}{32} \, \sqrt {2} \log \left (\frac {4 \, {\left (\sqrt {2} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} + x - \sqrt {x + 1} \sqrt {-x + 1} - 1\right )}}{x - 1}\right ) + \frac {3}{32} \, \sqrt {2} \log \left (-\frac {4 \, {\left (\sqrt {2} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - x + \sqrt {x + 1} \sqrt {-x + 1} + 1\right )}}{x - 1}\right ) \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*} \int \frac {{\left (x + 1\right )}^{\frac {1}{4}} x^{2}}{{\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.56, size = 451, normalized size = 1.93 \begin {gather*} \frac {\left (8 x^{2}+10 x +11\right ) \left (x +1\right )^{\frac {1}{4}} \left (x -1\right ) \left (\left (-x +1\right ) \left (x +1\right )^{3}\right )^{\frac {1}{4}}}{24 \left (-\left (x -1\right ) \left (x +1\right )^{3}\right )^{\frac {1}{4}} \left (-x +1\right )^{\frac {1}{4}}}+\frac {\left (\frac {3 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2} \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+x^{3}-\sqrt {-x^{4}-2 x^{3}+2 x +1}\, x \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 x^{2}-\sqrt {-x^{4}-2 x^{3}+2 x +1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+x +\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )}{\left (x +1\right )^{2}}\right )}{16}+\frac {3 \RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {x^{3}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2} \RootOf \left (\textit {\_Z}^{4}+1\right )+\sqrt {-x^{4}-2 x^{3}+2 x +1}\, x \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )^{3}+2 x^{2}+2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x \RootOf \left (\textit {\_Z}^{4}+1\right )+\sqrt {-x^{4}-2 x^{3}+2 x +1}\, \RootOf \left (\textit {\_Z}^{4}+1\right )^{2}+x +\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+1\right )}{\left (x +1\right )^{2}}\right )}{16}\right ) \left (\left (-x +1\right ) \left (x +1\right )^{3}\right )^{\frac {1}{4}}}{\left (x +1\right )^{\frac {3}{4}} \left (-x +1\right )^{\frac {1}{4}}} \end {gather*}
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*} \int \frac {{\left (x + 1\right )}^{\frac {1}{4}} x^{2}}{{\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \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 (x+1\right )}^{1/4}}{{\left (1-x\right )}^{1/4}} \,d x \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]{x + 1}}{\sqrt [4]{1 - x}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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