Optimal. Leaf size=156 \[ \frac {2}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )-\frac {2}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^5+x^3}}\right )-\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^5+x^3}}{\sqrt {x^5+x^3}-x^2}\right )+\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^5+x^3}}{\sqrt {2}}}{x \sqrt [4]{x^5+x^3}}\right ) \]
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Rubi [C] time = 0.74, antiderivative size = 320, normalized size of antiderivative = 2.05, number of steps used = 18, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2056, 6725, 959, 466, 510} \begin {gather*} -\frac {4 \left (1-\sqrt [3]{-1}\right ) \sqrt [4]{x^5+x^3} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^2,-\sqrt [3]{-1} x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {4 \left (1+(-1)^{2/3}\right ) \sqrt [4]{x^5+x^3} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^2,(-1)^{2/3} x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {8 \sqrt [4]{x^5+x^3} F_1\left (\frac {3}{8};1,-\frac {1}{4};\frac {11}{8};x^2,-x^2\right )}{9 \sqrt [4]{x^2+1}}-\frac {4 \left (1+(-1)^{2/3}\right ) x \sqrt [4]{x^5+x^3} F_1\left (\frac {7}{8};-\frac {1}{4},1;\frac {15}{8};-x^2,-\sqrt [3]{-1} x^2\right )}{21 \sqrt [4]{x^2+1}}-\frac {4 \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^5+x^3} F_1\left (\frac {7}{8};-\frac {1}{4},1;\frac {15}{8};-x^2,(-1)^{2/3} x^2\right )}{21 \sqrt [4]{x^2+1}}-\frac {8 x \sqrt [4]{x^5+x^3} F_1\left (\frac {7}{8};1,-\frac {1}{4};\frac {15}{8};x^2,-x^2\right )}{21 \sqrt [4]{x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 466
Rule 510
Rule 959
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx &=\frac {\sqrt [4]{x^3+x^5} \int \frac {(1+x) \sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (-1+x^3\right )} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^3+x^5} \int \left (-\frac {2 \sqrt [4]{1+x^2}}{3 (1-x) \sqrt [4]{x}}-\frac {\left (1+(-1)^{2/3}\right ) \sqrt [4]{1+x^2}}{3 \sqrt [4]{x} \left (1+\sqrt [3]{-1} x\right )}-\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt [4]{1+x^2}}{3 \sqrt [4]{x} \left (1-(-1)^{2/3} x\right )}\right ) \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=-\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{(1-x) \sqrt [4]{x}} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-(-1)^{2/3} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1+\sqrt [3]{-1} x\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}\\ &=-\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (2 \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1-x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1+\sqrt [3]{-1} x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left ((-1)^{2/3} \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1+\sqrt [3]{-1} x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {\sqrt [4]{1+x^2}}{\sqrt [4]{x} \left (1-(-1)^{2/3} x^2\right )} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (\sqrt [3]{-1} \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \int \frac {x^{3/4} \sqrt [4]{1+x^2}}{1-(-1)^{2/3} x^2} \, dx}{3 x^{3/4} \sqrt [4]{1+x^2}}\\ &=-\frac {\left (8 \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (8 \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1-x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1+\sqrt [3]{-1} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 (-1)^{2/3} \left (-1+\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1+\sqrt [3]{-1} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}+\frac {\left (4 \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^8}}{1-(-1)^{2/3} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}-\frac {\left (4 \sqrt [3]{-1} \left (-1-(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5}\right ) \operatorname {Subst}\left (\int \frac {x^6 \sqrt [4]{1+x^8}}{1-(-1)^{2/3} x^8} \, dx,x,\sqrt [4]{x}\right )}{3 x^{3/4} \sqrt [4]{1+x^2}}\\ &=-\frac {4 \left (1-\sqrt [3]{-1}\right ) \sqrt [4]{x^3+x^5} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^2,-\sqrt [3]{-1} x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {4 \left (1+(-1)^{2/3}\right ) \sqrt [4]{x^3+x^5} F_1\left (\frac {3}{8};-\frac {1}{4},1;\frac {11}{8};-x^2,(-1)^{2/3} x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {8 \sqrt [4]{x^3+x^5} F_1\left (\frac {3}{8};1,-\frac {1}{4};\frac {11}{8};x^2,-x^2\right )}{9 \sqrt [4]{1+x^2}}-\frac {4 (-1)^{2/3} \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^3+x^5} F_1\left (\frac {7}{8};-\frac {1}{4},1;\frac {15}{8};-x^2,-\sqrt [3]{-1} x^2\right )}{21 \sqrt [4]{1+x^2}}-\frac {4 \left (1-\sqrt [3]{-1}\right ) x \sqrt [4]{x^3+x^5} F_1\left (\frac {7}{8};-\frac {1}{4},1;\frac {15}{8};-x^2,(-1)^{2/3} x^2\right )}{21 \sqrt [4]{1+x^2}}-\frac {8 x \sqrt [4]{x^3+x^5} F_1\left (\frac {7}{8};1,-\frac {1}{4};\frac {15}{8};x^2,-x^2\right )}{21 \sqrt [4]{1+x^2}}\\ \end {align*}
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Mathematica [F] time = 1.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.53, size = 156, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt [4]{2} \tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )-\frac {1}{3} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{x^3+x^5}}{-x^2+\sqrt {x^3+x^5}}\right )-\frac {2}{3} \sqrt [4]{2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )+\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^3+x^5}}{\sqrt {2}}}{x \sqrt [4]{x^3+x^5}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 5.06, size = 953, normalized size = 6.11
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 + 1\right )}}{{\left (x^{3} - 1\right )} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 12.56, size = 732, normalized size = 4.69
method | result | size |
trager | \(-\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 )}{3}+\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 )}{3}+\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{5}+x^{3}}\, x +\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} x^{2}}{x^{2} \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}}{6}+\frac {\ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{5}+x^{3}}\, x +\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}+4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}} x^{2}}{x^{2} \left (x^{2}+x +1\right )}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{6}-\frac {\RootOf \left (\textit {\_Z}^{4}-2\right ) \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 )^{3} \left (x^{5}+x^{3}\right )^{\frac {1}{4}} x^{2}-\RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{4}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{4}+2 \RootOf \left (\textit {\_Z}^{4}-2\right ) \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \sqrt {x^{5}+x^{3}}\, x +\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{3}-2 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \sqrt {x^{5}+x^{3}}\, x +\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 ) x^{2}-\RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{x^{2} \left (x^{2}+x +1\right )}\right )}{3}\) | \(732\) |
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 + 1\right )}}{{\left (x^{3} - 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 = -0.01 \begin {gather*} \int \frac {{\left (x^5+x^3\right )}^{1/4}\,\left (x+1\right )}{x\,\left (x^3-1\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 + 1\right )}{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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