Optimal. Leaf size=255 \[ \frac {1}{8} \sqrt [4]{x^4+x^3} (4 x+1)-\frac {29}{16} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )+\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4+x^3}}\right )+\sqrt {\frac {1}{5} \left (2 \sqrt {5}-2\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4+x^3}}\right )+\frac {29}{16} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )-\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4+x^3}}\right )-\sqrt {\frac {1}{5} \left (2 \sqrt {5}-2\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4+x^3}}\right ) \]
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Rubi [A] time = 0.45, antiderivative size = 416, normalized size of antiderivative = 1.63, number of steps used = 26, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {1593, 2056, 903, 50, 63, 331, 298, 203, 206, 905, 911, 93} \begin {gather*} \frac {1}{2} \sqrt [4]{x^4+x^3} x+\frac {1}{8} \sqrt [4]{x^4+x^3}-\frac {29 \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{16 \sqrt [4]{x+1} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} \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 {5} \sqrt [4]{x+1} x^{3/4}}+\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} \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 {5} \sqrt [4]{x+1} x^{3/4}}+\frac {29 \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{16 \sqrt [4]{x+1} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{3+\sqrt {5}} \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 {5} \sqrt [4]{x+1} x^{3/4}}-\frac {2^{3/4} \sqrt [4]{3-\sqrt {5}} \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 {5} \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 203
Rule 206
Rule 298
Rule 331
Rule 903
Rule 905
Rule 911
Rule 1593
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (x+x^2\right ) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx &=\int \frac {x (1+x) \sqrt [4]{x^3+x^4}}{-1+x+x^2} \, dx\\ &=\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{7/4} (1+x)^{5/4}}{-1+x+x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int x^{3/4} \sqrt [4]{1+x} \, dx}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4} \sqrt [4]{1+x}}{-1+x+x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{2} x \sqrt [4]{x^3+x^4}+\frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4}}{(1+x)^{3/4}} \, dx}{8 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (-1+x+x^2\right )} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}-\frac {\left (3 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{32 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \left (-\frac {2}{\sqrt {5} \left (-1+\sqrt {5}-2 x\right ) \sqrt [4]{x} (1+x)^{3/4}}-\frac {2}{\sqrt {5} \sqrt [4]{x} (1+x)^{3/4} \left (1+\sqrt {5}+2 x\right )}\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^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}-\frac {\left (3 \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 )}{8 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 (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\left (-1+\sqrt {5}-2 x\right ) \sqrt [4]{x} (1+x)^{3/4}} \, dx}{\sqrt {5} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (1+\sqrt {5}+2 x\right )} \, dx}{\sqrt {5} x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}-\frac {\left (3 \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 )}{8 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 (8 \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 {5} x^{3/4} \sqrt [4]{1+x}}-\frac {\left (8 \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 {5} x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^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}}-\frac {\left (3 \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 )}{16 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (3 \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 )}{16 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt {\frac {2}{5}} \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 ) x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \sqrt {\frac {2}{5}} \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 ) x^{3/4} \sqrt [4]{1+x}}+\frac {\left (4 \sqrt {\frac {2}{5}} \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 ) x^{3/4} \sqrt [4]{1+x}}-\frac {\left (4 \sqrt {\frac {2}{5}} \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 ) x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {1}{8} \sqrt [4]{x^3+x^4}+\frac {1}{2} x \sqrt [4]{x^3+x^4}-\frac {29 \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{16 x^{3/4} \sqrt [4]{1+x}}+\frac {2^{3/4} \sqrt [4]{3+\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 {5} x^{3/4} \sqrt [4]{1+x}}+\frac {2^{3/4} \sqrt [4]{3-\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 {5} x^{3/4} \sqrt [4]{1+x}}+\frac {29 \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{16 x^{3/4} \sqrt [4]{1+x}}-\frac {2^{3/4} \sqrt [4]{3+\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 {5} x^{3/4} \sqrt [4]{1+x}}-\frac {2^{3/4} \sqrt [4]{3-\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 {5} x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}
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Mathematica [C] time = 0.43, size = 186, normalized size = 0.73 \begin {gather*} \frac {4}{15} \sqrt [4]{x^3 (x+1)} \left (\frac {5 \, _2F_1\left (-\frac {5}{4},\frac {3}{4};\frac {7}{4};-x\right )}{\sqrt [4]{x+1}}-\frac {5 \, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};-x\right )}{\sqrt [4]{x+1}}+\frac {5 \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )}{\sqrt [4]{x+1}}-\frac {2 \sqrt {5} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (-1+\sqrt {5}\right ) x}{\left (1+\sqrt {5}\right ) (x+1)}\right )}{\left (1+\sqrt {5}\right ) (x+1)}-\frac {2 \sqrt {5} \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {\left (1+\sqrt {5}\right ) x}{\left (-1+\sqrt {5}\right ) (x+1)}\right )}{\left (\sqrt {5}-1\right ) (x+1)}\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.95, size = 255, normalized size = 1.00 \begin {gather*} \frac {1}{8} (1+4 x) \sqrt [4]{x^3+x^4}-\frac {29}{16} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )+\sqrt {\frac {1}{5} \left (-2+2 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )+\frac {29}{16} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\sqrt {\frac {1}{5} \left (2+2 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^3+x^4}}\right )-\sqrt {\frac {1}{5} \left (-2+2 \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 = 0.52, size = 438, normalized size = 1.72 \begin {gather*} \frac {2}{5} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \arctan \left (\frac {\sqrt {2} x \sqrt {2 \, \sqrt {5} - 2} \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 {2 \, \sqrt {5} - 2}}{4 \, x}\right ) + \frac {2}{5} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \arctan \left (\frac {\sqrt {2} x \sqrt {2 \, \sqrt {5} + 2} \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 {2 \, \sqrt {5} + 2}}{4 \, x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (\frac {{\left (\sqrt {5} x - x\right )} \sqrt {2 \, \sqrt {5} + 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} + 2} \log \left (-\frac {{\left (\sqrt {5} x - x\right )} \sqrt {2 \, \sqrt {5} + 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (\frac {{\left (\sqrt {5} x + x\right )} \sqrt {2 \, \sqrt {5} - 2} + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{10} \, \sqrt {5} \sqrt {2 \, \sqrt {5} - 2} \log \left (-\frac {{\left (\sqrt {5} x + x\right )} \sqrt {2 \, \sqrt {5} - 2} - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{8} \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x + 1\right )} + \frac {29}{16} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {29}{32} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {29}{32} \, \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 = 1.12, size = 238, normalized size = 0.93 \begin {gather*} \frac {1}{8} \, {\left ({\left (\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} - \frac {1}{5} \, \sqrt {10 \, \sqrt {5} - 10} \arctan \left (\frac {{\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}}}\right ) - \frac {1}{5} \, \sqrt {10 \, \sqrt {5} + 10} \arctan \left (\frac {{\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}}{\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}}}\right ) - \frac {1}{10} \, \sqrt {10 \, \sqrt {5} - 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} + \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{10} \, \sqrt {10 \, \sqrt {5} + 10} \log \left (\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{10} \, \sqrt {10 \, \sqrt {5} - 10} \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}{10} \, \sqrt {10 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {\frac {1}{2} \, \sqrt {5} - \frac {1}{2}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + \frac {29}{16} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {29}{32} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {29}{32} \, \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 = 26.69, size = 2150, normalized size = 8.43
method | result | size |
trager | \(\text {Expression too large to display}\) | \(2150\) |
risch | \(\text {Expression too large to display}\) | \(4380\) |
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} + x\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 (x^2+x\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 {x \sqrt [4]{x^{3} \left (x + 1\right )} \left (x + 1\right )}{x^{2} + x - 1}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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