Optimal. Leaf size=164 \[ \frac {1}{7} x \left (x^4+x^2+1\right )^{3/2}+\frac {2}{21} x \left (3 x^2+4\right ) \sqrt {x^4+x^2+1}+\frac {2 x \sqrt {x^4+x^2+1}}{3 \left (x^2+1\right )}+\frac {4 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{7 \sqrt {x^4+x^2+1}}-\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{3 \sqrt {x^4+x^2+1}} \]
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Rubi [A] time = 0.06, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1206, 1176, 1197, 1103, 1195} \[ \frac {1}{7} x \left (x^4+x^2+1\right )^{3/2}+\frac {2}{21} x \left (3 x^2+4\right ) \sqrt {x^4+x^2+1}+\frac {2 x \sqrt {x^4+x^2+1}}{3 \left (x^2+1\right )}+\frac {4 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{7 \sqrt {x^4+x^2+1}}-\frac {2 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{3 \sqrt {x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1176
Rule 1195
Rule 1197
Rule 1206
Rubi steps
\begin {align*} \int \left (1+x^2\right )^2 \sqrt {1+x^2+x^4} \, dx &=\frac {1}{7} x \left (1+x^2+x^4\right )^{3/2}+\frac {1}{7} \int \left (6+10 x^2\right ) \sqrt {1+x^2+x^4} \, dx\\ &=\frac {2}{21} x \left (4+3 x^2\right ) \sqrt {1+x^2+x^4}+\frac {1}{7} x \left (1+x^2+x^4\right )^{3/2}+\frac {1}{105} \int \frac {50+70 x^2}{\sqrt {1+x^2+x^4}} \, dx\\ &=\frac {2}{21} x \left (4+3 x^2\right ) \sqrt {1+x^2+x^4}+\frac {1}{7} x \left (1+x^2+x^4\right )^{3/2}-\frac {2}{3} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx+\frac {8}{7} \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx\\ &=\frac {2 x \sqrt {1+x^2+x^4}}{3 \left (1+x^2\right )}+\frac {2}{21} x \left (4+3 x^2\right ) \sqrt {1+x^2+x^4}+\frac {1}{7} x \left (1+x^2+x^4\right )^{3/2}-\frac {2 \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 )}{3 \sqrt {1+x^2+x^4}}+\frac {4 \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 )}{7 \sqrt {1+x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 162, normalized size = 0.99 \[ \frac {2 \sqrt [3]{-1} \left (5 \sqrt [3]{-1}-7\right ) \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+14 \sqrt [3]{-1} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+x \left (3 x^8+12 x^6+23 x^4+20 x^2+11\right )}{21 \sqrt {x^4+x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (x^{4} + 2 \, x^{2} + 1\right )} \sqrt {x^{4} + x^{2} + 1}, x\right ) \]
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 \[ \int \sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 248, normalized size = 1.51 \[ \frac {\sqrt {x^{4}+x^{2}+1}\, x^{5}}{7}+\frac {3 \sqrt {x^{4}+x^{2}+1}\, x^{3}}{7}+\frac {11 \sqrt {x^{4}+x^{2}+1}\, x}{21}+\frac {20 \sqrt {-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{21 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {8 \sqrt {-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )+\EllipticF \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{3 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )} \]
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 \[ \int \sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}^{2}\,{d x} \]
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 \[ \int {\left (x^2+1\right )}^2\,\sqrt {x^4+x^2+1} \,d x \]
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 \[ \int \sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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