Optimal. Leaf size=48 \[ \sqrt [4]{x^4+x^3}+\frac {7}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right )-\frac {7}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4+x^3}}\right ) \]
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Rubi [A] time = 0.16, antiderivative size = 94, normalized size of antiderivative = 1.96, number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2056, 80, 63, 331, 298, 203, 206} \begin {gather*} \sqrt [4]{x^4+x^3}+\frac {7 \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 x^{3/4} \sqrt [4]{x+1}}-\frac {7 \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{2 x^{3/4} \sqrt [4]{x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 80
Rule 203
Rule 206
Rule 298
Rule 331
Rule 2056
Rubi steps
\begin {align*} \int \frac {(-1+x) \sqrt [4]{x^3+x^4}}{x (1+x)} \, dx &=\frac {\sqrt [4]{x^3+x^4} \int \frac {-1+x}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\sqrt [4]{x^3+x^4}-\frac {\left (7 \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{4 x^{3/4} \sqrt [4]{1+x}}\\ &=\sqrt [4]{x^3+x^4}-\frac {\left (7 \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}}\\ &=\sqrt [4]{x^3+x^4}-\frac {\left (7 \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}}\\ &=\sqrt [4]{x^3+x^4}-\frac {\left (7 \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 )}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (7 \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 )}{2 x^{3/4} \sqrt [4]{1+x}}\\ &=\sqrt [4]{x^3+x^4}+\frac {7 \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4} \sqrt [4]{1+x}}-\frac {7 \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{2 x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 45, normalized size = 0.94 \begin {gather*} \frac {x^3 \left (-7 (x+1)^{3/4} \, _2F_1\left (\frac {3}{4},\frac {3}{4};\frac {7}{4};-x\right )+3 x+3\right )}{3 \left (x^3 (x+1)\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.24, size = 48, normalized size = 1.00 \begin {gather*} \sqrt [4]{x^3+x^4}+\frac {7}{2} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {7}{2} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 65, normalized size = 1.35 \begin {gather*} {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}} - \frac {7}{2} \, \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {7}{4} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {7}{4} \, \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 = 0.39, size = 45, normalized size = 0.94 \begin {gather*} x {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} - \frac {7}{2} \, \arctan \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {7}{4} \, \log \left ({\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {7}{4} \, \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 = 0.76, size = 30, normalized size = 0.62
method | result | size |
meijerg | \(-\frac {4 \hypergeom \left (\left [\frac {3}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], -x \right ) x^{\frac {3}{4}}}{3}+\frac {4 \hypergeom \left (\left [\frac {3}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], -x \right ) x^{\frac {7}{4}}}{7}\) | \(30\) |
trager | \(\left (x^{4}+x^{3}\right )^{\frac {1}{4}}+\frac {7 \ln \left (\frac {2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}+x^{3}}\, x +2 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} x^{2}-2 x^{3}-x^{2}}{x^{2}}\right )}{4}-\frac {7 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}+x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}+x^{3}\right )^{\frac {3}{4}}-2 \left (x^{4}+x^{3}\right )^{\frac {1}{4}} x^{2}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}}{x^{2}}\right )}{4}\) | \(145\) |
risch | \(\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}+\frac {\left (-\frac {7 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x -\RootOf \left (\textit {\_Z}^{2}+1\right )+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}}{\left (1+x \right )^{2}}\right )}{4}-\frac {7 \ln \left (\frac {2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {3}{4}}+2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, x +2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x^{2}+2 x^{3}+2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}+4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} x +5 x^{2}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}}+4 x +1}{\left (1+x \right )^{2}}\right )}{4}\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (\left (1+x \right )^{3} x \right )^{\frac {1}{4}}}{x \left (1+x \right )}\) | \(369\) |
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 - 1\right )}}{{\left (x + 1\right )} x}\,{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 = -0.02 \begin {gather*} \int \frac {{\left (x^4+x^3\right )}^{1/4}\,\left (x-1\right )}{x\,\left (x+1\right )} \,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 )}{x \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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