Optimal. Leaf size=137 \[ -\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{x+1}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{x+1}}{192 x}-\frac {11}{64} \tan ^{-1}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {11}{64} \tanh ^{-1}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {99, 151, 12, 93, 212, 206, 203} \[ -\frac {29 (1-x)^{3/4} \sqrt [4]{x+1}}{96 x^2}-\frac {7 (1-x)^{3/4} \sqrt [4]{x+1}}{24 x^3}-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x^4}-\frac {83 (1-x)^{3/4} \sqrt [4]{x+1}}{192 x}-\frac {11}{64} \tan ^{-1}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {11}{64} \tanh ^{-1}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 93
Rule 99
Rule 151
Rule 203
Rule 206
Rule 212
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^5} \, dx &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}+\frac {1}{4} \int \frac {\frac {7}{2}+3 x}{\sqrt [4]{1-x} x^4 (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {1}{12} \int \frac {-\frac {29}{4}-7 x}{\sqrt [4]{1-x} x^3 (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}+\frac {1}{24} \int \frac {\frac {83}{8}+\frac {29 x}{4}}{\sqrt [4]{1-x} x^2 (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}-\frac {1}{24} \int -\frac {33}{16 \sqrt [4]{1-x} x (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}+\frac {11}{128} \int \frac {1}{\sqrt [4]{1-x} x (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}+\frac {11}{32} \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^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}-\frac {11}{64} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {11}{64} \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^4}-\frac {7 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x^3}-\frac {29 (1-x)^{3/4} \sqrt [4]{1+x}}{96 x^2}-\frac {83 (1-x)^{3/4} \sqrt [4]{1+x}}{192 x}-\frac {11}{64} \tan ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {11}{64} \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 = 67, normalized size = 0.49 \[ -\frac {(1-x)^{3/4} \left (22 x^4 \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {1-x}{x+1}\right )+83 x^4+141 x^3+114 x^2+104 x+48\right )}{192 x^4 (x+1)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 117, normalized size = 0.85 \[ \frac {66 \, x^{4} \arctan \left (\frac {{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + 33 \, x^{4} \log \left (\frac {x + {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 33 \, x^{4} \log \left (-\frac {x - {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 2 \, {\left (83 \, x^{3} + 58 \, x^{2} + 56 \, x + 48\right )} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{384 \, x^{4}} \]
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 \[ \int \frac {{\left (x + 1\right )}^{\frac {1}{4}}}{x^{5} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.65, size = 398, normalized size = 2.91 \[ \frac {\left (x +1\right )^{\frac {1}{4}} \left (x -1\right ) \left (83 x^{3}+58 x^{2}+56 x +48\right ) \left (\left (-x +1\right ) \left (x +1\right )^{3}\right )^{\frac {1}{4}}}{192 \left (-\left (x -1\right ) \left (x +1\right )^{3}\right )^{\frac {1}{4}} \left (-x +1\right )^{\frac {1}{4}} x^{4}}+\frac {\left (-\frac {11 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {x^{2} \RootOf \left (\textit {\_Z}^{2}+1\right )-\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}-\sqrt {-x^{4}-2 x^{3}+2 x +1}\, x \RootOf \left (\textit {\_Z}^{2}+1\right )+2 x \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x -\sqrt {-x^{4}-2 x^{3}+2 x +1}\, \RootOf \left (\textit {\_Z}^{2}+1\right )+\RootOf \left (\textit {\_Z}^{2}+1\right )+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}-\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}}{\left (x +1\right )^{2} x}\right )}{128}+\frac {11 \ln \left (\frac {\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}-x^{2}-\sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x -2 x +\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}-\sqrt {-x^{4}-2 x^{3}+2 x +1}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}-1}{\left (x +1\right )^{2} x}\right )}{128}\right ) \left (\left (-x +1\right ) \left (x +1\right )^{3}\right )^{\frac {1}{4}}}{\left (x +1\right )^{\frac {3}{4}} \left (-x +1\right )^{\frac {1}{4}}} \]
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 \[ \int \frac {{\left (x + 1\right )}^{\frac {1}{4}}}{x^{5} {\left (-x + 1\right )}^{\frac {1}{4}}}\,{d x} \]
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 \[ \int \frac {{\left (x+1\right )}^{1/4}}{x^5\,{\left (1-x\right )}^{1/4}} \,d x \]
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 \[ \int \frac {\sqrt [4]{x + 1}}{x^{5} \sqrt [4]{1 - x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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