Optimal. Leaf size=68 \[ \frac {4 \sqrt [4]{x^4+x^3}}{x}+\sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 114, normalized size of antiderivative = 1.68, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2056, 848, 96, 93, 298, 203, 206} \begin {gather*} \frac {4 \sqrt [4]{x^4+x^3}}{x}+\frac {\sqrt [4]{2} \sqrt [4]{x^4+x^3} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{2} \sqrt [4]{x^4+x^3} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 93
Rule 96
Rule 203
Rule 206
Rule 298
Rule 848
Rule 2056
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{x^3+x^4}}{x^2 \left (-1+x^2\right )} \, dx &=\frac {\sqrt [4]{x^3+x^4} \int \frac {\sqrt [4]{1+x}}{x^{5/4} \left (-1+x^2\right )} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {\sqrt [4]{x^3+x^4} \int \frac {1}{(-1+x) x^{5/4} (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {4 \sqrt [4]{x^3+x^4}}{x}+\frac {\sqrt [4]{x^3+x^4} \int \frac {1}{(-1+x) \sqrt [4]{x} (1+x)^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {4 \sqrt [4]{x^3+x^4}}{x}+\frac {\left (4 \sqrt [4]{x^3+x^4}\right ) \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 {4 \sqrt [4]{x^3+x^4}}{x}-\frac {\left (\sqrt {2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\sqrt {2} \sqrt [4]{x^3+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ &=\frac {4 \sqrt [4]{x^3+x^4}}{x}+\frac {\sqrt [4]{2} \sqrt [4]{x^3+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{2} \sqrt [4]{x^3+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 42, normalized size = 0.62 \begin {gather*} -\frac {4 x^2 \left (x \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};\frac {2 x}{x+1}\right )-3 (x+1)\right )}{3 \left (x^3 (x+1)\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.34, size = 68, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{x^4+x^3}}{x}+\sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 128, normalized size = 1.88 \begin {gather*} \frac {4 \cdot 8^{\frac {3}{4}} x \arctan \left (\frac {8^{\frac {1}{4}} x \sqrt {\frac {\sqrt {2} x^{2} + \sqrt {x^{4} + x^{3}}}{x^{2}}} - 8^{\frac {1}{4}} {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{2 \, x}\right ) - 8^{\frac {3}{4}} x \log \left (\frac {8^{\frac {3}{4}} x + 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 8^{\frac {3}{4}} x \log \left (-\frac {8^{\frac {3}{4}} x - 4 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 32 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{8 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.20, size = 65, normalized size = 0.96 \begin {gather*} -2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{2} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \right |}\right ) + 4 \, {\left (\frac {1}{x} + 1\right )}^{\frac {1}{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 1.54, size = 541, normalized size = 7.96 \begin {gather*} \frac {4 \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}+\frac {\left (-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x +2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}+2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x +3 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{3}+4 \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}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}+7 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}+5 \RootOf \left (\textit {\_Z}^{4}-2\right ) x +\RootOf \left (\textit {\_Z}^{4}-2\right )}{\left (-1+x \right ) \left (1+x \right )^{2}}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x +2 \sqrt {x^{4}+3 x^{3}+3 x^{2}+x}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}-2 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{3}-4 \left (x^{4}+3 x^{3}+3 x^{2}+x \right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x -7 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}+4 \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}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}-5 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (-1+x \right ) \left (1+x \right )^{2}}\right )}{2}\right ) \left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}} \left (x \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (1+x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {{\left (x^4+x^3\right )}^{1/4}}{x^2-x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^{3} \left (x + 1\right )}}{x^{2} \left (x - 1\right ) \left (x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________