Optimal. Leaf size=190 \[ \frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{2\ 2^{3/4}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )}{2\ 2^{3/4}}-\frac {3 \tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^5+x^3}}{\sqrt {2} x^2-\sqrt {x^5+x^3}}\right )}{4 \sqrt [4]{2}}-\frac {3 \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^5+x^3}}{2^{3/4}}}{x \sqrt [4]{x^5+x^3}}\right )}{4 \sqrt [4]{2}}+\frac {4 \sqrt [4]{x^5+x^3} \left (x^4+2 x^2+1\right )}{9 x^3} \]
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Rubi [C] time = 0.61, antiderivative size = 211, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2056, 1586, 6725, 364, 466, 510} \begin {gather*} \frac {4 \sqrt [4]{x^5+x^3} F_1\left (-\frac {9}{8};1,\frac {3}{4};-\frac {1}{8};x^2,-x^2\right )}{3 \sqrt [4]{x^2+1} x^3}+\frac {4 \sqrt [4]{x^5+x^3} x^3 \, _2F_1\left (\frac {3}{4},\frac {15}{8};\frac {23}{8};-x^2\right )}{15 \sqrt [4]{x^2+1}}+\frac {4 \sqrt [4]{x^5+x^3} x \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};-x^2\right )}{7 \sqrt [4]{x^2+1}}-\frac {8 \sqrt [4]{x^5+x^3} \, _2F_1\left (-\frac {1}{8},\frac {3}{4};\frac {7}{8};-x^2\right )}{\sqrt [4]{x^2+1} x}-\frac {8 \sqrt [4]{x^5+x^3} \, _2F_1\left (-\frac {9}{8},\frac {3}{4};-\frac {1}{8};-x^2\right )}{9 \sqrt [4]{x^2+1} x^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 364
Rule 466
Rule 510
Rule 1586
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx &=\frac {\sqrt [4]{x^3+x^5} \int \frac {\sqrt [4]{1+x^2} \left (1+x^4+x^8\right )}{x^{13/4} \left (-1+x^4\right )} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^3+x^5} \int \frac {1+x^4+x^8}{x^{13/4} \left (-1+x^2\right ) \left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^3+x^5} \int \left (\frac {2}{x^{13/4} \left (1+x^2\right )^{3/4}}+\frac {2}{x^{5/4} \left (1+x^2\right )^{3/4}}+\frac {x^{3/4}}{\left (1+x^2\right )^{3/4}}+\frac {x^{11/4}}{\left (1+x^2\right )^{3/4}}+\frac {3}{x^{13/4} \left (-1+x^2\right ) \left (1+x^2\right )^{3/4}}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^3+x^5} \int \frac {x^{3/4}}{\left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^3+x^5} \int \frac {x^{11/4}}{\left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {1}{x^{13/4} \left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {1}{x^{5/4} \left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (3 \sqrt [4]{x^3+x^5}\right ) \int \frac {1}{x^{13/4} \left (-1+x^2\right ) \left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=-\frac {8 \sqrt [4]{x^3+x^5} \, _2F_1\left (-\frac {9}{8},\frac {3}{4};-\frac {1}{8};-x^2\right )}{9 x^3 \sqrt [4]{1+x^2}}-\frac {8 \sqrt [4]{x^3+x^5} \, _2F_1\left (-\frac {1}{8},\frac {3}{4};\frac {7}{8};-x^2\right )}{x \sqrt [4]{1+x^2}}+\frac {4 x \sqrt [4]{x^3+x^5} \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};-x^2\right )}{7 \sqrt [4]{1+x^2}}+\frac {4 x^3 \sqrt [4]{x^3+x^5} \, _2F_1\left (\frac {3}{4},\frac {15}{8};\frac {23}{8};-x^2\right )}{15 \sqrt [4]{1+x^2}}+\frac {\left (12 \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{10} \left (-1+x^8\right ) \left (1+x^8\right )^{3/4}} \, 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 {9}{8};1,\frac {3}{4};-\frac {1}{8};x^2,-x^2\right )}{3 x^3 \sqrt [4]{1+x^2}}-\frac {8 \sqrt [4]{x^3+x^5} \, _2F_1\left (-\frac {9}{8},\frac {3}{4};-\frac {1}{8};-x^2\right )}{9 x^3 \sqrt [4]{1+x^2}}-\frac {8 \sqrt [4]{x^3+x^5} \, _2F_1\left (-\frac {1}{8},\frac {3}{4};\frac {7}{8};-x^2\right )}{x \sqrt [4]{1+x^2}}+\frac {4 x \sqrt [4]{x^3+x^5} \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};-x^2\right )}{7 \sqrt [4]{1+x^2}}+\frac {4 x^3 \sqrt [4]{x^3+x^5} \, _2F_1\left (\frac {3}{4},\frac {15}{8};\frac {23}{8};-x^2\right )}{15 \sqrt [4]{1+x^2}}\\ \end {align*}
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Mathematica [F] time = 0.55, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x^3+x^5} \left (1+x^4+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.66, size = 190, normalized size = 1.00 \begin {gather*} \frac {4 \left (1+2 x^2+x^4\right ) \sqrt [4]{x^3+x^5}}{9 x^3}+\frac {3 \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{2\ 2^{3/4}}-\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 \sqrt [4]{2}}-\frac {3 \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )}{2\ 2^{3/4}}-\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 \sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 6.71, size = 1133, normalized size = 5.96
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^{8} + x^{4} + 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}}{{\left (x^{4} - 1\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 29.58, size = 742, normalized size = 3.91
method | result | size |
trager | \(\frac {4 \left (x^{4}+2 x^{2}+1\right ) \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{9 x^{3}}+\frac {3 \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}+4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{3} 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}}}{\left (1+x \right )^{2} x^{2}}\right )}{8}-\frac {3 \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}-4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right ) 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}}}{\left (1+x \right )^{2} x^{2}}\right )}{8}-\frac {3 \ln \left (-\frac {4 \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}^{4}+2\right )^{2} 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}}{x^{2} \left (-1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{3}}{16}-\frac {3 \ln \left (-\frac {4 \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}^{4}+2\right )^{2} 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}}{x^{2} \left (-1+x \right )^{2}}\right ) \RootOf \left (\textit {\_Z}^{4}+2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+2\right )^{2}\right )}{16}+\frac {3 \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}}{x^{2} \left (-1+x \right )^{2}}\right )}{8}\) | \(742\) |
risch | \(\text {Expression too large to display}\) | \(1784\) |
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^{8} + x^{4} + 1\right )} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}}{{\left (x^{4} - 1\right )} x^{4}}\,{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^8+x^4+1\right )}{x^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 {\sqrt [4]{x^{3} \left (x^{2} + 1\right )} \left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}{x^{4} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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