Optimal. Leaf size=173 \[ -\frac {1}{4} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4+2\& ,\frac {-\text {$\#$1}^4 \log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )+\text {$\#$1}^4 \log (x)+2 \log \left (\sqrt [4]{x^4+x^3}-\text {$\#$1} x\right )-2 \log (x)}{\text {$\#$1}^7-\text {$\#$1}^3}\& \right ]-2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )}{2^{3/4}}+2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )}{2^{3/4}} \]
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Rubi [C] time = 1.99, antiderivative size = 616, normalized size of antiderivative = 3.56, number of steps used = 67, number of rules used = 21, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.954, Rules used = {2056, 1586, 6733, 6742, 331, 298, 203, 206, 1240, 410, 237, 335, 275, 231, 407, 409, 1213, 537, 494, 6725, 1529} \begin {gather*} -\frac {2 \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 {(1-i)^{5/4} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{x+1}}+\frac {(1+i)^{5/4} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2^{3/4} x^{3/4} \sqrt [4]{x+1}}+\frac {2 \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 {(1-i)^{5/4} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{x+1}}-\frac {(1+i)^{5/4} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{4 x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2^{3/4} x^{3/4} \sqrt [4]{x+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 231
Rule 237
Rule 275
Rule 298
Rule 331
Rule 335
Rule 407
Rule 409
Rule 410
Rule 494
Rule 537
Rule 1213
Rule 1240
Rule 1529
Rule 1586
Rule 2056
Rule 6725
Rule 6733
Rule 6742
Rubi steps
\begin {align*} \int \frac {x^2 \sqrt [4]{x^3+x^4}}{-1+x^4} \, dx &=\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{11/4} \sqrt [4]{1+x}}{-1+x^4} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{11/4}}{(1+x)^{3/4} \left (-1+x-x^2+x^3\right )} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\left (1+x^4\right )^{3/4} \left (-1+x^4-x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{\left (1+x^4\right )^{3/4}}+\frac {x^2 \left (1-x^4+x^8\right )}{\left (1+x^4\right )^{3/4} \left (-1+x^4-x^8+x^{12}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\left (4 \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 (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (1-x^4+x^8\right )}{\left (1+x^4\right )^{3/4} \left (-1+x^4-x^8+x^{12}\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\left (4 \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 (4 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{4 \left (-1+x^2\right ) \left (1+x^4\right )^{3/4}}+\frac {1}{4 \left (1+x^2\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2 \left (-1+x^4\right )}{2 \left (1+x^4\right )^{3/4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \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 (2 \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 (2 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^4\right )}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {2 \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 {2 \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 {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \left (\frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \left (\frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {x^2}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )}+\frac {x^6}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {2 \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 {2 \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 {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^4\right )^{3/4} \left (1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {2 \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 {2 \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 {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{1-x^4} \, dx,x,\sqrt [4]{x}\right )}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{-1+x^4} \, dx,x,\sqrt [4]{x}\right )}{2 x^{3/4} \sqrt [4]{1+x}}+2 \frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (-\frac {i x^2}{2 \left (-i+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {i x^2}{2 \left (i+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {x^2}{2 \left (-i+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{2 \left (i+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {2 \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 {2 \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 (i \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (i \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{\left (-i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{\left (i+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}+2 \left (-\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2} x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 \sqrt {2} x^{3/4} \sqrt [4]{1+x}}\right )+\frac {\left (\sqrt {\frac {1}{1+x}} \sqrt [4]{1+x} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-2 x^4\right ) \sqrt {1-x^4}} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4}}+\frac {\left (\sqrt {\frac {1}{1+x}} \sqrt [4]{1+x} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^4} \left (-1+2 x^4\right )} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4}}\\ &=-\frac {2 \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 {2 \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}}+2 \left (\frac {\sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}\right )-\frac {\left (i \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{i+(1-i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (i \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{-i+(1+i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{i+(1-i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \operatorname {Subst}\left (\int \frac {x^2}{-i+(1+i) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {2 \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 {2 \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}}+2 \left (\frac {\sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}\right )-\frac {\left (i (1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (i (1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\left ((1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((1-i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {1-i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (i (1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\left (i (1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\left ((1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\left ((1+i)^{3/2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {1+i} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}\\ &=-\frac {2 \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 {\sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{1+x}}+\frac {(1-i)^{5/4} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{1+x}}+\frac {(1+i)^{5/4} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+\frac {2 \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 {\sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1-i)^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {(1-i)^{5/4} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1-i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 (1+i)^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {(1+i)^{5/4} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{1+i} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{4 x^{3/4} \sqrt [4]{1+x}}+2 \left (\frac {\sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2\ 2^{3/4} x^{3/4} \sqrt [4]{1+x}}\right )\\ \end {align*}
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Mathematica [C] time = 0.22, size = 221, normalized size = 1.28 \begin {gather*} \frac {\sqrt [4]{x^3 (x+1)} \left (-7 \left (11 x^4 \, _2F_1\left (\frac {3}{4},\frac {15}{4};\frac {19}{4};-x\right )+11 x^3 \, _2F_1\left (\frac {3}{4},\frac {15}{4};\frac {19}{4};-x\right )-15 (x+1) x^2 \, _2F_1\left (-\frac {1}{4},\frac {11}{4};\frac {15}{4};-x\right )-165 (x+1) \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )+(55-55 i) \sqrt [4]{x+1} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {(1-i) x}{x+1}\right )+(55+55 i) \sqrt [4]{x+1} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {(1+i) x}{x+1}\right )+110 \sqrt [4]{x+1} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {2 x}{x+1}\right )\right )+385 (x+1) \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-x\right )-165 x (x+1) \, _2F_1\left (-\frac {1}{4},\frac {7}{4};\frac {11}{4};-x\right )\right )}{1155 (x+1)^{5/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.43, size = 173, normalized size = 1.00 \begin {gather*} -2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )}{2^{3/4}}+2 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^4}}\right )}{2^{3/4}}-\frac {1}{4} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 \log (x)+2 \log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^4-\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}^3+\text {$\#$1}^7}\&\right ] \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 2015, normalized size = 11.65
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.29, size = 257, normalized size = 1.49
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 13.62, size = 1932, normalized size = 11.17
method | result | size |
trager | \(\text {Expression too large to display}\) | \(1932\) |
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}} x^{2}}{x^{4} - 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 = -0.01 \begin {gather*} \int \frac {x^2\,{\left (x^4+x^3\right )}^{1/4}}{x^4-1} \,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 {x^{2} \sqrt [4]{x^{3} \left (x + 1\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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