Optimal. Leaf size=183 \[ \frac {1}{3} \left (x^4+x^2+1\right )^{3/2} x+\frac {2}{45} \left (6 x^2+7\right ) \sqrt {x^4+x^2+1} x+\frac {26 \sqrt {x^4+x^2+1} x}{45 \left (x^2+1\right )}+\frac {7 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{15 \sqrt {x^4+x^2+1}}-\frac {26 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{45 \sqrt {x^4+x^2+1}}+\frac {1}{9} \left (x^4+x^2+1\right )^{3/2} x^3 \]
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Rubi [A] time = 0.09, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {1206, 1679, 1176, 1197, 1103, 1195} \[ \frac {1}{9} \left (x^4+x^2+1\right )^{3/2} x^3+\frac {1}{3} \left (x^4+x^2+1\right )^{3/2} x+\frac {2}{45} \left (6 x^2+7\right ) \sqrt {x^4+x^2+1} x+\frac {26 \sqrt {x^4+x^2+1} x}{45 \left (x^2+1\right )}+\frac {7 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{15 \sqrt {x^4+x^2+1}}-\frac {26 \left (x^2+1\right ) \sqrt {\frac {x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{4}\right )}{45 \sqrt {x^4+x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1176
Rule 1195
Rule 1197
Rule 1206
Rule 1679
Rubi steps
\begin {align*} \int \left (1+x^2\right )^3 \sqrt {1+x^2+x^4} \, dx &=\frac {1}{9} x^3 \left (1+x^2+x^4\right )^{3/2}+\frac {1}{9} \int \sqrt {1+x^2+x^4} \left (9+24 x^2+21 x^4\right ) \, dx\\ &=\frac {1}{3} x \left (1+x^2+x^4\right )^{3/2}+\frac {1}{9} x^3 \left (1+x^2+x^4\right )^{3/2}+\frac {1}{63} \int \left (42+84 x^2\right ) \sqrt {1+x^2+x^4} \, dx\\ &=\frac {2}{45} x \left (7+6 x^2\right ) \sqrt {1+x^2+x^4}+\frac {1}{3} x \left (1+x^2+x^4\right )^{3/2}+\frac {1}{9} x^3 \left (1+x^2+x^4\right )^{3/2}+\frac {1}{945} \int \frac {336+546 x^2}{\sqrt {1+x^2+x^4}} \, dx\\ &=\frac {2}{45} x \left (7+6 x^2\right ) \sqrt {1+x^2+x^4}+\frac {1}{3} x \left (1+x^2+x^4\right )^{3/2}+\frac {1}{9} x^3 \left (1+x^2+x^4\right )^{3/2}-\frac {26}{45} \int \frac {1-x^2}{\sqrt {1+x^2+x^4}} \, dx+\frac {14}{15} \int \frac {1}{\sqrt {1+x^2+x^4}} \, dx\\ &=\frac {26 x \sqrt {1+x^2+x^4}}{45 \left (1+x^2\right )}+\frac {2}{45} x \left (7+6 x^2\right ) \sqrt {1+x^2+x^4}+\frac {1}{3} x \left (1+x^2+x^4\right )^{3/2}+\frac {1}{9} x^3 \left (1+x^2+x^4\right )^{3/2}-\frac {26 \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 )}{45 \sqrt {1+x^2+x^4}}+\frac {7 \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 )}{15 \sqrt {1+x^2+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.32, size = 169, normalized size = 0.92 \[ \frac {2 (-1)^{5/6} \left (4 \sqrt {3}+9 i\right ) \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+26 \sqrt [3]{-1} \sqrt {\sqrt [3]{-1} x^2+1} \sqrt {1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+x \left (5 x^{10}+25 x^8+57 x^6+81 x^4+61 x^2+29\right )}{45 \sqrt {x^4+x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (x^{6} + 3 \, x^{4} + 3 \, x^{2} + 1\right )} \sqrt {x^{4} + x^{2} + 1}, x\right ) \]
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 \[ \int \sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.16, size = 263, normalized size = 1.44 \[ \frac {\sqrt {x^{4}+x^{2}+1}\, x^{7}}{9}+\frac {4 \sqrt {x^{4}+x^{2}+1}\, x^{5}}{9}+\frac {32 \sqrt {x^{4}+x^{2}+1}\, x^{3}}{45}+\frac {29 \sqrt {x^{4}+x^{2}+1}\, x}{45}+\frac {32 \sqrt {-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \EllipticF \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )}{45 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}}-\frac {104 \sqrt {-\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \sqrt {-\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )+\EllipticF \left (\frac {\sqrt {-2+2 i \sqrt {3}}\, x}{2}, \frac {\sqrt {-2+2 i \sqrt {3}}}{2}\right )\right )}{45 \sqrt {-2+2 i \sqrt {3}}\, \sqrt {x^{4}+x^{2}+1}\, \left (1+i \sqrt {3}\right )} \]
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 \[ \int \sqrt {x^{4} + x^{2} + 1} {\left (x^{2} + 1\right )}^{3}\,{d x} \]
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 \[ \int {\left (x^2+1\right )}^3\,\sqrt {x^4+x^2+1} \,d x \]
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 \[ \int \sqrt {\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )} \left (x^{2} + 1\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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