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 = 0.155488, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 11, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.55, Rules used = {90, 80, 50, 63, 331, 297, 1162, 617, 204, 1165, 628} \[ -\frac{1}{3} (1-x)^{3/4} x (x+1)^{5/4}-\frac{1}{12} (1-x)^{3/4} (x+1)^{5/4}-\frac{3}{8} (1-x)^{3/4} \sqrt [4]{x+1}-\frac{3 \log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{16 \sqrt{2}}+\frac{3 \log \left (\frac{\sqrt{1-x}}{\sqrt{x+1}}+\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{16 \sqrt{2}}+\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}\right )}{8 \sqrt{2}}-\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{1-x}}{\sqrt [4]{x+1}}+1\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 90
Rule 80
Rule 50
Rule 63
Rule 331
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
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*}
Mathematica [C] time = 0.0247893, size = 58, normalized size = 0.25 \[ -\frac{1}{12} (1-x)^{3/4} \left (6 \sqrt [4]{2} \, _2F_1\left (-\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac{1-x}{2}\right )+\sqrt [4]{x+1} \left (4 x^2+5 x+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.023, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2}\sqrt [4]{1+x}{\frac{1}{\sqrt [4]{1-x}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x + 1\right )}^{\frac{1}{4}} x^{2}}{{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67995, size = 853, normalized size = 3.65 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2} \sqrt [4]{x + 1}}{\sqrt [4]{1 - x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (x + 1\right )}^{\frac{1}{4}} x^{2}}{{\left (-x + 1\right )}^{\frac{1}{4}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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