Optimal. Leaf size=145 \[ \frac {1}{5} \left (x^2+2\right ) \sqrt {x^4+x^2+1} x+\frac {3 \sqrt {x^4+x^2+1} x}{5 \left (x^2+1\right )}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{5 \sqrt {x^4+x^2+1}}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{5 \sqrt {x^4+x^2+1}} \]
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Rubi [A] time = 0.04, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1176, 1197, 1103, 1195} \[ \frac {1}{5} \left (x^2+2\right ) \sqrt {x^4+x^2+1} x+\frac {3 \sqrt {x^4+x^2+1} x}{5 \left (x^2+1\right )}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{5 \sqrt {x^4+x^2+1}}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{5 \sqrt {x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1176
Rule 1195
Rule 1197
Rubi steps
\begin {align*} \int \left (1+x^2\right ) \sqrt {1+x^2+x^4} \, dx &=\frac {1}{5} x \left (2+x^2\right ) \sqrt {1+x^2+x^4}+\frac {1}{15} \int \frac {9+9 x^2}{\sqrt {1+x^2+x^4}} \, dx\\ &=\frac {1}{5} x \left (2+x^2\right ) \sqrt {1+x^2+x^4}-\frac {3}{5} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx+\frac {6}{5} \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx\\ &=\frac {3 x \sqrt {1+x^2+x^4}}{5 \left (1+x^2\right )}+\frac {1}{5} x \left (2+x^2\right ) \sqrt {1+x^2+x^4}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{5 \sqrt {1+x^2+x^4}}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{5 \sqrt {1+x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 168, normalized size = 1.16 \[ \frac {x^7+3 x^5+3 x^3+\frac {3}{2} \sqrt {2+\left (1-i \sqrt {3}\right ) x^2} \sqrt {2+\left (1+i \sqrt {3}\right ) x^2} F\left (\sin ^{-1}\left (\frac {1}{2} \left (i \sqrt {3} x+x\right )\right )|\frac {1}{2} i \left (i+\sqrt {3}\right )\right )+3 \sqrt [3]{-1} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+2 x}{5 \sqrt {x^4+x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}, x\right ) \]
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 \[ \int \sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.00, size = 233, normalized size = 1.61 \[ \frac {\sqrt {x^{4}+x^{2}+1}\, x^{3}}{5}+\frac {2 \sqrt {x^{4}+x^{2}+1}\, x}{5}+\frac {6 \sqrt {-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{5 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {12 \sqrt {-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )+\EllipticF \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{5 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )} \]
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 \[ \int \sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}\,{d x} \]
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 \[ \int \left (x^2+1\right )\,\sqrt {x^4+x^2+1} \,d x \]
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 \[ \int \sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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