Optimal. Leaf size=46 \[ x \sqrt {-x^4+x^2+2} \left (x^2+2\right )+3 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+7 E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {1176, 1180, 524, 424, 419} \[ x \sqrt {-x^4+x^2+2} \left (x^2+2\right )+3 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+7 E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 1176
Rule 1180
Rubi steps
\begin {align*} \int \left (7+5 x^2\right ) \sqrt {2+x^2-x^4} \, dx &=x \left (2+x^2\right ) \sqrt {2+x^2-x^4}-\frac {1}{15} \int \frac {-150-105 x^2}{\sqrt {2+x^2-x^4}} \, dx\\ &=x \left (2+x^2\right ) \sqrt {2+x^2-x^4}-\frac {2}{15} \int \frac {-150-105 x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx\\ &=x \left (2+x^2\right ) \sqrt {2+x^2-x^4}+6 \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx+7 \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx\\ &=x \left (2+x^2\right ) \sqrt {2+x^2-x^4}+7 E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+3 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\\ \end {align*}
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Mathematica [C] time = 0.08, size = 94, normalized size = 2.04 \[ \frac {-x^7-x^5+4 x^3-12 i \sqrt {-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+7 i \sqrt {-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+4 x}{\sqrt {-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}, x\right ) \]
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 \[ \int \sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 141, normalized size = 3.07 \[ \sqrt {-x^{4}+x^{2}+2}\, x^{3}+2 \sqrt {-x^{4}+x^{2}+2}\, x +\frac {5 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{\sqrt {-x^{4}+x^{2}+2}}-\frac {7 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )+\EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )\right )}{2 \sqrt {-x^{4}+x^{2}+2}} \]
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 \[ \int \sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \left (5\,x^2+7\right )\,\sqrt {-x^4+x^2+2} \,d x \]
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 \[ \int \sqrt {- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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