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 = 0.019335, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, 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 96
Rule 94
Rule 93
Rule 212
Rule 206
Rule 203
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*}
Mathematica [C] time = 0.0144115, 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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Maple [F] time = 0.021, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}}\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^{3}{\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 = 1.61223, size = 300, normalized size = 3.3 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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^{3}{\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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