Optimal. Leaf size=261 \[ \frac {1}{4} \sqrt [4]{x^4-x^3} (4 x+11)-\frac {49}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )+\sqrt {2 \left (11+5 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4-x^3}}\right )-\sqrt {2 \left (5 \sqrt {5}-11\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4-x^3}}\right )+\frac {49}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )-\sqrt {2 \left (11+5 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4-x^3}}\right )+\sqrt {2 \left (5 \sqrt {5}-11\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4-x^3}}\right ) \]
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Rubi [B] time = 1.15, antiderivative size = 672, normalized size of antiderivative = 2.57, number of steps used = 34, number of rules used = 12, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2056, 6728, 50, 63, 240, 212, 206, 203, 101, 157, 93, 298} \begin {gather*} -\sqrt [4]{x^4-x^3} (1-x)+\frac {1}{2} \left (3+\sqrt {5}\right ) \sqrt [4]{x^4-x^3}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{x^4-x^3}+\frac {3}{4} \sqrt [4]{x^4-x^3}+\frac {\left (13+7 \sqrt {5}\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (13-7 \sqrt {5}\right ) \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}-\frac {3 \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (3+\sqrt {5}\right )^{5/4} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{2} \sqrt [4]{x-1} x^{3/4}}-\frac {\left (3-\sqrt {5}\right )^{5/4} \sqrt [4]{x^4-x^3} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{2} \sqrt [4]{x-1} x^{3/4}}+\frac {\left (13+7 \sqrt {5}\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}+\frac {\left (13-7 \sqrt {5}\right ) \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{x-1} x^{3/4}}-\frac {3 \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{x-1} x^{3/4}}-\frac {\left (3+\sqrt {5}\right )^{5/4} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{2} \sqrt [4]{x-1} x^{3/4}}+\frac {\left (3-\sqrt {5}\right )^{5/4} \sqrt [4]{x^4-x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{x-1}}\right )}{\sqrt [4]{2} \sqrt [4]{x-1} x^{3/4}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 50
Rule 63
Rule 93
Rule 101
Rule 157
Rule 203
Rule 206
Rule 212
Rule 240
Rule 298
Rule 2056
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x+2 x^2\right ) \sqrt [4]{-x^3+x^4}}{-1-x+x^2} \, dx &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} \left (-1+x+2 x^2\right )}{-1-x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \left (2 \sqrt [4]{-1+x} x^{3/4}+\frac {\sqrt [4]{-1+x} x^{3/4} (1+3 x)}{-1-x+x^2}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {\sqrt [4]{-x^3+x^4} \int \frac {\sqrt [4]{-1+x} x^{3/4} (1+3 x)}{-1-x+x^2} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt [4]{-x^3+x^4}\right ) \int \sqrt [4]{-1+x} x^{3/4} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=-\left ((1-x) \sqrt [4]{-x^3+x^4}\right )+\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}} \, dx}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\sqrt [4]{-x^3+x^4} \int \left (\frac {\left (3+\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}{-1-\sqrt {5}+2 x}+\frac {\left (3-\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}{-1+\sqrt {5}+2 x}\right ) \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{16 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-1+\sqrt {5}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\sqrt [4]{-1+x} x^{3/4}}{-1-\sqrt {5}+2 x} \, dx}{\sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{4} \left (1-\sqrt {5}\right )+\frac {1}{2} \left (-1+2 \sqrt {5}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1+\sqrt {5}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {\frac {3}{4} \left (1+\sqrt {5}\right )+\frac {1}{2} \left (-1-2 \sqrt {5}\right ) x}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-\sqrt {5}+2 x\right )} \, dx}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1+\sqrt {5}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (\left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x} \left (-1-\sqrt {5}+2 x\right )} \, dx}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (-1-2 \sqrt {5}\right ) \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3-\sqrt {5}\right ) \left (-1+2 \sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{8 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (3 \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1+\sqrt {5}-\left (1+\sqrt {5}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {\left (4 \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-1-\sqrt {5}-\left (1-\sqrt {5}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (-1-2 \sqrt {5}\right ) \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3-\sqrt {5}\right ) \left (-1+2 \sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {3 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {3 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 \sqrt {2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\left (-1-\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt {2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\left (-1-\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (-1-2 \sqrt {5}\right ) \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (2 \sqrt {2} \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\left (1-\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (2 \sqrt {2} \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\left (1-\sqrt {5}\right ) \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3-\sqrt {5}\right ) \left (-1+2 \sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{2 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {3 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{123+55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{123-55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {3 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{123+55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{123-55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (-1-2 \sqrt {5}\right ) \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (-1-2 \sqrt {5}\right ) \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3-\sqrt {5}\right ) \left (-1+2 \sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {\left (\left (3-\sqrt {5}\right ) \left (-1+2 \sqrt {5}\right ) \sqrt [4]{-x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}\\ &=\frac {3}{4} \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3-\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}+\frac {1}{2} \left (3+\sqrt {5}\right ) \sqrt [4]{-x^3+x^4}-(1-x) \sqrt [4]{-x^3+x^4}-\frac {3 \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (13-7 \sqrt {5}\right ) \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (13+7 \sqrt {5}\right ) \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{123+55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{123-55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}-\frac {3 \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (13-7 \sqrt {5}\right ) \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}+\frac {\left (13+7 \sqrt {5}\right ) \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{4 \sqrt [4]{-1+x} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{123+55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{123-55 \sqrt {5}} \sqrt [4]{-x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \sqrt [4]{x}}{\sqrt [4]{-1+x}}\right )}{\sqrt [4]{-1+x} x^{3/4}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 168, normalized size = 0.64 \begin {gather*} \frac {4 \sqrt [4]{(x-1) x^3} \left (15 \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {1}{4};\frac {5}{4};1-x\right )+2 (x-1) \sqrt [4]{x} \, _2F_1\left (-\frac {3}{4},\frac {5}{4};\frac {9}{4};1-x\right )+5 \sqrt [4]{x} \, _2F_1\left (\frac {1}{4},\frac {1}{4};\frac {5}{4};1-x\right )+5 \left (\sqrt {5}-2\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {\left (-1+\sqrt {5}\right ) (x-1)}{\left (1+\sqrt {5}\right ) x}\right )-5 \left (2+\sqrt {5}\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};\frac {\left (1+\sqrt {5}\right ) (x-1)}{\left (-1+\sqrt {5}\right ) x}\right )\right )}{5 x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 1.47, size = 261, normalized size = 1.00 \begin {gather*} \frac {1}{4} (11+4 x) \sqrt [4]{-x^3+x^4}-\frac {49}{8} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )+\sqrt {2 \left (11+5 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right )-\sqrt {2 \left (-11+5 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right )+\frac {49}{8} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\sqrt {2 \left (11+5 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right )+\sqrt {2 \left (-11+5 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-x^3+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 474, normalized size = 1.82 \begin {gather*} -2 \, \sqrt {10 \, \sqrt {5} - 22} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {5} x + 3 \, x\right )} \sqrt {10 \, \sqrt {5} - 22} \sqrt {\frac {\sqrt {5} x^{2} + x^{2} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {10 \, \sqrt {5} - 22} {\left (\sqrt {5} + 3\right )}}{8 \, x}\right ) - 2 \, \sqrt {10 \, \sqrt {5} + 22} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {5} x - 3 \, x\right )} \sqrt {10 \, \sqrt {5} + 22} \sqrt {\frac {\sqrt {5} x^{2} - x^{2} + 2 \, \sqrt {x^{4} - x^{3}}}{x^{2}}} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} \sqrt {10 \, \sqrt {5} + 22} {\left (\sqrt {5} - 3\right )}}{8 \, x}\right ) - \frac {1}{2} \, \sqrt {10 \, \sqrt {5} + 22} \log \left (\frac {{\left (\sqrt {5} x - 2 \, x\right )} \sqrt {10 \, \sqrt {5} + 22} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {10 \, \sqrt {5} + 22} \log \left (-\frac {{\left (\sqrt {5} x - 2 \, x\right )} \sqrt {10 \, \sqrt {5} + 22} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \sqrt {10 \, \sqrt {5} - 22} \log \left (\frac {{\left (\sqrt {5} x + 2 \, x\right )} \sqrt {10 \, \sqrt {5} - 22} + 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \sqrt {10 \, \sqrt {5} - 22} \log \left (-\frac {{\left (\sqrt {5} x + 2 \, x\right )} \sqrt {10 \, \sqrt {5} - 22} - 2 \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{4} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x + 11\right )} + \frac {49}{8} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {49}{16} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {49}{16} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.10, size = 261, normalized size = 1.00 \begin {gather*} \frac {1}{4} \, {\left (11 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - 15 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} - \sqrt {10 \, \sqrt {5} - 22} \arctan \left (\frac {{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) + \sqrt {10 \, \sqrt {5} + 22} \arctan \left (\frac {{\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{2} \, \sqrt {10 \, \sqrt {5} - 22} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \sqrt {10 \, \sqrt {5} + 22} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \, \sqrt {10 \, \sqrt {5} - 22} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {1}{2} \, \sqrt {10 \, \sqrt {5} + 22} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) - \frac {49}{8} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {49}{16} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {49}{16} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 22.60, size = 1909, normalized size = 7.31
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1909\) |
risch | \(\text {Expression too large to display}\) | \(3971\) |
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^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (2 \, x^{2} + x - 1\right )}}{x^{2} - x - 1}\,{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 {{\left (x^4-x^3\right )}^{1/4}\,\left (2\,x^2+x-1\right )}{-x^2+x+1} \,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 {\sqrt [4]{x^{3} \left (x - 1\right )} \left (x + 1\right ) \left (2 x - 1\right )}{x^{2} - x - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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