Optimal. Leaf size=184 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{4 \sqrt [4]{2}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{4 \sqrt [4]{2}}+\frac {\left (x^5+x^3\right )^{3/4}}{x^2 \left (x^2+1\right )}-\frac {3 \tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^5+x^3}}{\sqrt {2} x^2-\sqrt {x^5+x^3}}\right )}{4\ 2^{3/4}}+\frac {3 \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^5+x^3}}{2^{3/4}}}{x \sqrt [4]{x^5+x^3}}\right )}{4\ 2^{3/4}} \]
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Rubi [C] time = 0.65, antiderivative size = 94, normalized size of antiderivative = 0.51, number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2056, 6715, 6725, 245, 1455, 527, 530, 429} \begin {gather*} \frac {6 \sqrt [4]{x^2+1} x F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};x^2,-x^2\right )}{\sqrt [4]{x^5+x^3}}-\frac {3 \sqrt [4]{x^2+1} x \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^2\right )}{\sqrt [4]{x^5+x^3}}+\frac {x}{\sqrt [4]{x^5+x^3}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 245
Rule 429
Rule 527
Rule 530
Rule 1455
Rule 2056
Rule 6715
Rule 6725
Rubi steps
\begin {align*} \int \frac {1+x^2+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx &=\frac {\left (x^{3/4} \sqrt [4]{1+x^2}\right ) \int \frac {1+x^2+x^4}{x^{3/4} \sqrt [4]{1+x^2} \left (1-x^4\right )} \, dx}{\sqrt [4]{x^3+x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1+x^8+x^{16}}{\sqrt [4]{1+x^8} \left (1-x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}\\ &=\frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{\sqrt [4]{1+x^8}}+\frac {2+x^8}{\sqrt [4]{1+x^8} \left (1-x^{16}\right )}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}\\ &=-\frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}+\frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2+x^8}{\sqrt [4]{1+x^8} \left (1-x^{16}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}\\ &=-\frac {4 x \sqrt [4]{1+x^2} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^2\right )}{\sqrt [4]{x^3+x^5}}+\frac {\left (4 x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {2+x^8}{\left (1-x^8\right ) \left (1+x^8\right )^{5/4}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}\\ &=\frac {x}{\sqrt [4]{x^3+x^5}}-\frac {4 x \sqrt [4]{1+x^2} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^2\right )}{\sqrt [4]{x^3+x^5}}-\frac {\left (x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {-7+x^8}{\left (1-x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}\\ &=\frac {x}{\sqrt [4]{x^3+x^5}}-\frac {4 x \sqrt [4]{1+x^2} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^2\right )}{\sqrt [4]{x^3+x^5}}+\frac {\left (x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}+\frac {\left (6 x^{3/4} \sqrt [4]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-x^8\right ) \sqrt [4]{1+x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{x^3+x^5}}\\ &=\frac {x}{\sqrt [4]{x^3+x^5}}+\frac {6 x \sqrt [4]{1+x^2} F_1\left (\frac {1}{8};1,\frac {1}{4};\frac {9}{8};x^2,-x^2\right )}{\sqrt [4]{x^3+x^5}}-\frac {3 x \sqrt [4]{1+x^2} \, _2F_1\left (\frac {1}{8},\frac {1}{4};\frac {9}{8};-x^2\right )}{\sqrt [4]{x^3+x^5}}\\ \end {align*}
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Mathematica [F] time = 0.58, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1+x^2+x^4}{\left (1-x^4\right ) \sqrt [4]{x^3+x^5}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.62, size = 184, normalized size = 1.00 \begin {gather*} \frac {\left (x^3+x^5\right )^{3/4}}{x^2 \left (1+x^2\right )}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{4 \sqrt [4]{2}}-\frac {3 \tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^3+x^5}}{\sqrt {2} x^2-\sqrt {x^3+x^5}}\right )}{4\ 2^{3/4}}+\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{4 \sqrt [4]{2}}+\frac {3 \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 )}{4\ 2^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 17.93, size = 1102, normalized size = 5.99
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 {x^{4} + x^{2} + 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 31.23, size = 733, normalized size = 3.98
method | result | size |
risch | \(\frac {x}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x -2 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}}{x^{2} \left (-1+x \right )^{2}}\right )}{16}+\frac {3 \RootOf \left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} \sqrt {x^{5}+x^{3}}\, x +2 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}+x^{4} \RootOf \left (\textit {\_Z}^{4}-8\right )+2 \RootOf \left (\textit {\_Z}^{4}-8\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}-8\right ) x^{2}}{x^{2} \left (-1+x \right )^{2}}\right )}{16}-\frac {3 \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{4}-2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{2}-16 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-8\right ) x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{\left (1+x \right )^{2} x^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{3}}{64}-\frac {3 \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{4}-2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{2}-16 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-8\right ) x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{\left (1+x \right )^{2} x^{2}}\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2}}{64}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{4}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-8\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2}-\RootOf \left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{2} \RootOf \left (\textit {\_Z}^{4}-8\right )^{2}-8 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}-8\right ) x -8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-8\right )^{2}\right ) \sqrt {x^{5}+x^{3}}\, x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{\left (1+x \right )^{2} x^{2}}\right )}{32}\) | \(733\) |
trager | \(\frac {\left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (x^{2}+1\right )}+\frac {3 \RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} \sqrt {x^{5}+x^{3}}\, x -2 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{4}+8\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{4}+8\right ) x^{2}}{\left (1+x \right )^{2} x^{2}}\right )}{16}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {\sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x +2 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{3}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}}{\left (1+x \right )^{2} x^{2}}\right )}{16}+\frac {3 \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{4}-2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{2}-16 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{3}}{64}+\frac {3 \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{4}-2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{2}-16 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{64}-\frac {3 \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{4}-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{3}+8 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+8\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{2}-\RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}-8 \sqrt {x^{5}+x^{3}}\, \RootOf \left (\textit {\_Z}^{4}+8\right ) x -8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \sqrt {x^{5}+x^{3}}\, x +16 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (-1+x \right )^{2}}\right )}{32}\) | \(739\) |
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 {x^{4} + x^{2} + 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{4} - 1\right )}}\,{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^4+x^2+1}{{\left (x^5+x^3\right )}^{1/4}\,\left (x^4-1\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 {x^{2}}{x^{4} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx - \int \frac {x^{4}}{x^{4} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx - \int \frac {1}{x^{4} \sqrt [4]{x^{5} + x^{3}} - \sqrt [4]{x^{5} + x^{3}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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