Optimal. Leaf size=169 \[ \frac {4 \sqrt [4]{x^5+x^3}}{x}+\sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^5+x^3}}{\sqrt {2} x^2-\sqrt {x^5+x^3}}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^5+x^3}}{2^{3/4}}}{x \sqrt [4]{x^5+x^3}}\right )}{\sqrt [4]{2}} \]
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Rubi [C] time = 0.18, antiderivative size = 44, normalized size of antiderivative = 0.26, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2056, 466, 510} \begin {gather*} \frac {4 \sqrt [4]{x^5+x^3} F_1\left (-\frac {1}{8};1,-\frac {5}{4};\frac {7}{8};x^2,-x^2\right )}{x \sqrt [4]{x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 466
Rule 510
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (-1+x^2\right )} \, dx &=\frac {\sqrt [4]{x^3+x^5} \int \frac {\left (1+x^2\right )^{5/4}}{x^{5/4} \left (-1+x^2\right )} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {\left (4 \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {\left (1+x^8\right )^{5/4}}{x^2 \left (-1+x^8\right )} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {4 \sqrt [4]{x^3+x^5} F_1\left (-\frac {1}{8};1,-\frac {5}{4};\frac {7}{8};x^2,-x^2\right )}{x \sqrt [4]{1+x^2}}\\ \end {align*}
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Mathematica [F] time = 0.11, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (-1+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.43, size = 169, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{x^3+x^5}}{x}+\sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^3+x^5}}{\sqrt {2} x^2-\sqrt {x^3+x^5}}\right )}{\sqrt [4]{2}}-\sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^3+x^5}}{2^{3/4}}}{x \sqrt [4]{x^3+x^5}}\right )}{\sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 6.00, size = 1053, normalized size = 6.23
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^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}}{{\left (x^{2} - 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 28.95, size = 721, normalized size = 4.27
method | result | size |
trager | \(\frac {4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{x}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{5}+x^{3}\right )^{\frac {1}{4}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {x^{5}+x^{3}}\, x +4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+x \right )^{2}}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{3}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x^{2}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{5}+x^{3}\right )^{\frac {1}{4}} x^{2}+4 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-2\right ) x +4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+x \right )^{2}}\right )}{2}-\frac {\ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \sqrt {x^{5}+x^{3}}\, x +4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{5}+x^{3}\right )^{\frac {1}{4}} x^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}}{\left (1+x \right )^{2} x^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}}{4}-\frac {\ln \left (\frac {4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \sqrt {x^{5}+x^{3}}\, x +4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (x^{5}+x^{3}\right )^{\frac {1}{4}} x^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}}{\left (1+x \right )^{2} x^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (-\frac {-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{5}+x^{3}}\, x +2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} \sqrt {x^{5}+x^{3}}\, x -4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}+\RootOf \left (\textit {\_Z}^{4}-2\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}}{\left (1+x \right )^{2} x^{2}}\right )}{2}\) | \(721\) |
risch | \(\text {Expression too large to display}\) | \(1752\) |
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^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}}{{\left (x^{2} - 1\right )} x^{2}}\,{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 {{\left (x^5+x^3\right )}^{1/4}\,\left (x^2+1\right )}{x^2\,\left (x^2-1\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^{2} + 1\right )} \left (x^{2} + 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.
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