Optimal. Leaf size=91 \[ -\frac {(1-x)^{3/4} (x+1)^{5/4}}{2 x^2}-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x}-\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {96, 94, 93, 212, 206, 203} \[ -\frac {(1-x)^{3/4} (x+1)^{5/4}}{2 x^2}-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x}-\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]
Antiderivative was successfully verified.
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Rule 93
Rule 94
Rule 96
Rule 203
Rule 206
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^3} \, dx &=-\frac {(1-x)^{3/4} (1+x)^{5/4}}{2 x^2}+\frac {1}{4} \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^2} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x}-\frac {(1-x)^{3/4} (1+x)^{5/4}}{2 x^2}+\frac {1}{8} \int \frac {1}{\sqrt [4]{1-x} x (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x}-\frac {(1-x)^{3/4} (1+x)^{5/4}}{2 x^2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x}-\frac {(1-x)^{3/4} (1+x)^{5/4}}{2 x^2}-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x}-\frac {(1-x)^{3/4} (1+x)^{5/4}}{2 x^2}-\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 57, normalized size = 0.63 \[ -\frac {(1-x)^{3/4} \left (2 x^2 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {1-x}{x+1}\right )+9 x^2+15 x+6\right )}{12 x^2 (x+1)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.91, size = 106, normalized size = 1.16 \[ \frac {2 \, x^{2} \arctan \left (\frac {{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + x^{2} \log \left (\frac {x + {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - x^{2} \log \left (-\frac {x - {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 2 \, {\left (3 \, x + 2\right )} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{8 \, x^{2}} \]
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 \[ \int \frac {{\left (x + 1\right )}^{\frac {1}{4}}}{x^{3} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.81, size = 389, normalized size = 4.27 \[ \frac {\left (x +1\right )^{\frac {1}{4}} \left (x -1\right ) \left (3 x +2\right ) \left (\left (-x +1\right ) \left (x +1\right )^{3}\right )^{\frac {1}{4}}}{4 \left (-\left (x -1\right ) \left (x +1\right )^{3}\right )^{\frac {1}{4}} \left (-x +1\right )^{\frac {1}{4}} x^{2}}+\frac {\left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-x^{2} \RootOf \left (\textit {\_Z}^{2}+1\right )+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}+\sqrt {-x^{4}-2 x^{3}+2 x +1}\, x \RootOf \left (\textit {\_Z}^{2}+1\right )-2 x \RootOf \left (\textit {\_Z}^{2}+1\right )+2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x +\sqrt {-x^{4}-2 x^{3}+2 x +1}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-\RootOf \left (\textit {\_Z}^{2}+1\right )-\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}}{\left (x +1\right )^{2} x}\right )}{8}+\frac {\ln \left (\frac {\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}-x^{2}-\sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x -2 x +\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}-\sqrt {-x^{4}-2 x^{3}+2 x +1}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}-1}{\left (x +1\right )^{2} x}\right )}{8}\right ) \left (\left (-x +1\right ) \left (x +1\right )^{3}\right )^{\frac {1}{4}}}{\left (x +1\right )^{\frac {3}{4}} \left (-x +1\right )^{\frac {1}{4}}} \]
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 \[ \int \frac {{\left (x + 1\right )}^{\frac {1}{4}}}{x^{3} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (x+1\right )}^{1/4}}{x^3\,{\left (1-x\right )}^{1/4}} \,d x \]
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 \[ \int \frac {\sqrt [4]{x + 1}}{x^{3} \sqrt [4]{1 - x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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