Optimal. Leaf size=118 \[ \frac {2 \sqrt [4]{x^6+x^2}}{x}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^6+x^2}}{\sqrt {2} x^2-\sqrt {x^6+x^2}}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^6+x^2}}{2^{3/4}}}{x \sqrt [4]{x^6+x^2}}\right )}{\sqrt [4]{2}} \]
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Rubi [C] time = 0.59, antiderivative size = 147, normalized size of antiderivative = 1.25, number of steps used = 20, number of rules used = 11, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {2056, 6725, 277, 329, 364, 1312, 1336, 325, 1337, 466, 510} \begin {gather*} \frac {8 \sqrt [4]{x^6+x^2} x F_1\left (\frac {3}{8};1,\frac {3}{4};\frac {11}{8};x^4,-x^4\right )}{3 \sqrt [4]{x^4+1}}-\frac {8 \sqrt [4]{x^6+x^2} x^3 F_1\left (\frac {7}{8};1,\frac {3}{4};\frac {15}{8};x^4,-x^4\right )}{7 \sqrt [4]{x^4+1}}-\frac {4 \sqrt [4]{x^6+x^2} x \, _2F_1\left (\frac {3}{8},\frac {3}{4};\frac {11}{8};-x^4\right )}{3 \sqrt [4]{x^4+1}}+\frac {2 \sqrt [4]{x^6+x^2}}{x} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 277
Rule 325
Rule 329
Rule 364
Rule 466
Rule 510
Rule 1312
Rule 1336
Rule 1337
Rule 2056
Rule 6725
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx &=\frac {\sqrt [4]{x^2+x^6} \int \frac {\left (-1+x^2\right ) \sqrt [4]{1+x^4}}{x^{3/2} \left (1+x^2\right )} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\sqrt [4]{x^2+x^6} \int \left (\frac {\sqrt [4]{1+x^4}}{x^{3/2}}-\frac {2 \sqrt [4]{1+x^4}}{x^{3/2} \left (1+x^2\right )}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {\sqrt [4]{x^2+x^6} \int \frac {\sqrt [4]{1+x^4}}{x^{3/2}} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}-\frac {\left (2 \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt [4]{1+x^4}}{x^{3/2} \left (1+x^2\right )} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=-\frac {2 \sqrt [4]{x^2+x^6}}{x}+\frac {\left (2 \sqrt [4]{x^2+x^6}\right ) \int \frac {x^{5/2}}{\left (1+x^4\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}-\frac {\left (2 \sqrt [4]{x^2+x^6}\right ) \int \frac {1+x^2}{x^{3/2} \left (1+x^4\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (4 \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x}}{\left (1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=-\frac {2 \sqrt [4]{x^2+x^6}}{x}-\frac {\left (2 \sqrt [4]{x^2+x^6}\right ) \int \left (\frac {1}{x^{3/2} \left (1+x^4\right )^{3/4}}+\frac {\sqrt {x}}{\left (1+x^4\right )^{3/4}}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (4 \sqrt [4]{x^2+x^6}\right ) \int \left (\frac {\sqrt {x}}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^{5/2}}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (4 \sqrt [4]{x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=-\frac {2 \sqrt [4]{x^2+x^6}}{x}+\frac {4 x^3 \sqrt [4]{x^2+x^6} \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};-x^4\right )}{7 \sqrt [4]{1+x^4}}-\frac {\left (2 \sqrt [4]{x^2+x^6}\right ) \int \frac {1}{x^{3/2} \left (1+x^4\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}-\frac {\left (2 \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x}}{\left (1+x^4\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (4 \sqrt [4]{x^2+x^6}\right ) \int \frac {\sqrt {x}}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (4 \sqrt [4]{x^2+x^6}\right ) \int \frac {x^{5/2}}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {2 \sqrt [4]{x^2+x^6}}{x}+\frac {4 x^3 \sqrt [4]{x^2+x^6} \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};-x^4\right )}{7 \sqrt [4]{1+x^4}}-\frac {\left (2 \sqrt [4]{x^2+x^6}\right ) \int \frac {x^{5/2}}{\left (1+x^4\right )^{3/4}} \, dx}{\sqrt {x} \sqrt [4]{1+x^4}}-\frac {\left (4 \sqrt [4]{x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1+x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (8 \sqrt [4]{x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^8\right ) \left (1+x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}+\frac {\left (8 \sqrt [4]{x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (-1+x^8\right ) \left (1+x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {2 \sqrt [4]{x^2+x^6}}{x}+\frac {8 x \sqrt [4]{x^2+x^6} F_1\left (\frac {3}{8};1,\frac {3}{4};\frac {11}{8};x^4,-x^4\right )}{3 \sqrt [4]{1+x^4}}-\frac {8 x^3 \sqrt [4]{x^2+x^6} F_1\left (\frac {7}{8};1,\frac {3}{4};\frac {15}{8};x^4,-x^4\right )}{7 \sqrt [4]{1+x^4}}-\frac {4 x \sqrt [4]{x^2+x^6} \, _2F_1\left (\frac {3}{8},\frac {3}{4};\frac {11}{8};-x^4\right )}{3 \sqrt [4]{1+x^4}}+\frac {4 x^3 \sqrt [4]{x^2+x^6} \, _2F_1\left (\frac {3}{4},\frac {7}{8};\frac {15}{8};-x^4\right )}{7 \sqrt [4]{1+x^4}}-\frac {\left (4 \sqrt [4]{x^2+x^6}\right ) \operatorname {Subst}\left (\int \frac {x^6}{\left (1+x^8\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^4}}\\ &=\frac {2 \sqrt [4]{x^2+x^6}}{x}+\frac {8 x \sqrt [4]{x^2+x^6} F_1\left (\frac {3}{8};1,\frac {3}{4};\frac {11}{8};x^4,-x^4\right )}{3 \sqrt [4]{1+x^4}}-\frac {8 x^3 \sqrt [4]{x^2+x^6} F_1\left (\frac {7}{8};1,\frac {3}{4};\frac {15}{8};x^4,-x^4\right )}{7 \sqrt [4]{1+x^4}}-\frac {4 x \sqrt [4]{x^2+x^6} \, _2F_1\left (\frac {3}{8},\frac {3}{4};\frac {11}{8};-x^4\right )}{3 \sqrt [4]{1+x^4}}\\ \end {align*}
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Mathematica [F] time = 1.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.48, size = 118, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt [4]{x^6+x^2}}{x}-\frac {\tan ^{-1}\left (\frac {2^{3/4} x \sqrt [4]{x^6+x^2}}{\sqrt {2} x^2-\sqrt {x^6+x^2}}\right )}{\sqrt [4]{2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{\sqrt [4]{2}}+\frac {\sqrt {x^6+x^2}}{2^{3/4}}}{x \sqrt [4]{x^6+x^2}}\right )}{\sqrt [4]{2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 7.32, size = 545, normalized size = 4.62 \begin {gather*} -\frac {4 \cdot 2^{\frac {3}{4}} x \arctan \left (-\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} - 4 \, \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {x^{5} + 2 \, x^{3} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + 2 \, x^{3} + x}} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{2 \, {\left (x^{5} - 2 \, x^{3} + x\right )}}\right ) + 4 \cdot 2^{\frac {3}{4}} x \arctan \left (-\frac {4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} - \sqrt {2} {\left (2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + \sqrt {2} {\left (x^{5} + 2 \, x^{3} + x\right )} + 4 \, \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}\right )} \sqrt {\frac {x^{5} + 2 \, x^{3} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x - 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x}{x^{5} + 2 \, x^{3} + x}} + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{2 \, {\left (x^{5} - 2 \, x^{3} + x\right )}}\right ) + 2^{\frac {3}{4}} x \log \left (\frac {2 \, {\left (x^{5} + 2 \, x^{3} + 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x + 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + 2 \, x^{3} + x}\right ) - 2^{\frac {3}{4}} x \log \left (\frac {2 \, {\left (x^{5} + 2 \, x^{3} - 4 \cdot 2^{\frac {1}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} x^{2} + 4 \, \sqrt {2} \sqrt {x^{6} + x^{2}} x - 2 \cdot 2^{\frac {3}{4}} {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}} + x\right )}}{x^{5} + 2 \, x^{3} + x}\right ) - 16 \, {\left (x^{6} + x^{2}\right )}^{\frac {1}{4}}}{8 \, x} \end {gather*}
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^{6} + x^{2}\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 [F] time = 5.81, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{2}-1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{x^{2} \left (x^{2}+1\right )}\, dx \end {gather*}
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^{6} + x^{2}\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^6+x^2\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^{2} \left (x^{4} + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{x^{2} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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