Optimal. Leaf size=211 \[ -\frac {3}{2} \text {RootSum}\left [\text {$\#$1}^8-3 \text {$\#$1}^4+3\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+\log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )-\log (x)}{2 \text {$\#$1}^7-3 \text {$\#$1}^3}\& \right ]+\frac {1}{2} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)-\log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )+\log (x)}{2 \text {$\#$1}^7-\text {$\#$1}^3}\& \right ]-2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right ) \]
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Rubi [C] time = 2.82, antiderivative size = 1271, normalized size of antiderivative = 6.02, number of steps used = 35, number of rules used = 12, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {2056, 6728, 906, 63, 331, 298, 203, 206, 6725, 93, 205, 208} \begin {gather*} -\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{x+1}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{x+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 93
Rule 203
Rule 205
Rule 206
Rule 208
Rule 298
Rule 331
Rule 906
Rule 2056
Rule 6725
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^4}}{1+x^2+x^4} \, dx &=\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4} \sqrt [4]{1+x} \left (-1+x^2\right )}{1+x^2+x^4} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int \left (\frac {\left (1+i \sqrt {3}\right ) x^{3/4} \sqrt [4]{1+x}}{1-i \sqrt {3}+2 x^2}+\frac {\left (1-i \sqrt {3}\right ) x^{3/4} \sqrt [4]{1+x}}{1+i \sqrt {3}+2 x^2}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x}}{1+i \sqrt {3}+2 x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x}}{1-i \sqrt {3}+2 x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {-1-i \sqrt {3}+2 x}{\sqrt [4]{x} (1+x)^{3/4} \left (1+i \sqrt {3}+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {-1+i \sqrt {3}+2 x}{\sqrt [4]{x} (1+x)^{3/4} \left (1-i \sqrt {3}+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \left (\frac {\sqrt {2} \left (-1-i \sqrt {3}\right )+\left (-1-i \sqrt {3}\right )^{3/2}}{2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}-\sqrt {2} x\right )}+\frac {-\sqrt {2} \left (-1-i \sqrt {3}\right )+\left (-1-i \sqrt {3}\right )^{3/2}}{2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \int \left (\frac {\sqrt {2} \left (-1+i \sqrt {3}\right )+\left (-1+i \sqrt {3}\right )^{3/2}}{2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}-\sqrt {2} x\right )}+\frac {-\sqrt {2} \left (-1+i \sqrt {3}\right )+\left (-1+i \sqrt {3}\right )^{3/2}}{2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}+\sqrt {2} x\right )}\right ) \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\left (2 \left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}-\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1-i \sqrt {3}}+\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}-\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (\sqrt {-1+i \sqrt {3}}+\sqrt {2} x\right )} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1-i \sqrt {3}}-\left (\sqrt {2}+\sqrt {-1-i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1-i \sqrt {3}}-\left (-\sqrt {2}+\sqrt {-1-i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+i \sqrt {3}}-\left (\sqrt {2}+\sqrt {-1+i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+i \sqrt {3}}-\left (-\sqrt {2}+\sqrt {-1+i \sqrt {3}}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-i \sqrt {3}}-\sqrt {-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-i \sqrt {3}}+\sqrt {-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1-i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-i \sqrt {3}}-\sqrt {\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1-i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1-i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1-i \sqrt {3}}+\sqrt {\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {\sqrt {2}+\sqrt {-1-i \sqrt {3}}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+i \sqrt {3}}-\sqrt {-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+i \sqrt {3}}+\sqrt {-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (1+i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+i \sqrt {3}}-\sqrt {\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\left (1+i \sqrt {3}\right ) \left (-\sqrt {2}-\sqrt {-1+i \sqrt {3}}\right ) \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{-1+i \sqrt {3}}+\sqrt {\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {\sqrt {2}+\sqrt {-1+i \sqrt {3}}} x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1-i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1-i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1-i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{-\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{\sqrt {2}+\sqrt {-1+i \sqrt {3}}} \sqrt [4]{x}}{\sqrt [8]{-1+i \sqrt {3}} \sqrt [4]{1+x}}\right )}{2 \sqrt [8]{-1+i \sqrt {3}} x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}
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Mathematica [C] time = 0.80, size = 489, normalized size = 2.32 \begin {gather*} \frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3 (x+1)} \left (x^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )-\frac {\left (1-\frac {1}{\sqrt {\frac {1}{2} \left (-1-i \sqrt {3}\right )}}\right ) x^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (1-i \sqrt {3}\right ) x}{2 (x+1)}\right )}{(x+1)^{3/4}}\right )}{3 x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3 (x+1)} \left (x^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )-\frac {\left (1-\frac {1}{\sqrt {\frac {1}{2} i \left (\sqrt {3}+i\right )}}\right ) x^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (1+i \sqrt {3}\right ) x}{2 (x+1)}\right )}{(x+1)^{3/4}}\right )}{3 x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1-i \sqrt {3}\right ) \sqrt [4]{x^3 (x+1)} \left (2 x^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )-\frac {\left (3+i \sqrt {3}\right ) x^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (1+\frac {1}{\sqrt {\frac {1}{2} \left (-1-i \sqrt {3}\right )}}\right ) x}{x+1}\right )}{(x+1)^{3/4}}\right )}{6 x^{3/4} \sqrt [4]{x+1}}+\frac {\left (1+i \sqrt {3}\right ) \sqrt [4]{x^3 (x+1)} \left (2 x^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )-\frac {\left (3-i \sqrt {3}\right ) x^{3/4} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (1+\frac {1}{\sqrt {\frac {1}{2} i \left (i+\sqrt {3}\right )}}\right ) x}{x+1}\right )}{(x+1)^{3/4}}\right )}{6 x^{3/4} \sqrt [4]{x+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.00, size = 211, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {3}{2} \text {RootSum}\left [3-3 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-3 \text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ]+\frac {1}{2} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {\log (x)-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+2 \text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.96, size = 3011, normalized size = 14.27
result too large to display
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 \begin {gather*} \int \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{x^{4} + x^{2} + 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 211.98, size = 14158, normalized size = 67.10
method | result | size |
trager | \(\text {Expression too large to display}\) | \(14158\) |
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 (x^{2} - 1\right )}}{x^{4} + x^{2} + 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 (x^2-1\right )}{x^4+x^2+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 (x + 1\right )}{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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