Optimal. Leaf size=74 \[ -\frac {25}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac {1}{21} x \left (354 x^2+275\right ) \sqrt {-x^4+x^2+2}-\frac {79}{7} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {2045}{21} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1206, 1176, 1180, 524, 424, 419} \[ -\frac {25}{7} x \left (-x^4+x^2+2\right )^{3/2}+\frac {1}{21} x \left (354 x^2+275\right ) \sqrt {-x^4+x^2+2}-\frac {79}{7} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {2045}{21} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 1176
Rule 1180
Rule 1206
Rubi steps
\begin {align*} \int \left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4} \, dx &=-\frac {25}{7} x \left (2+x^2-x^4\right )^{3/2}-\frac {1}{7} \int \left (-393-590 x^2\right ) \sqrt {2+x^2-x^4} \, dx\\ &=\frac {1}{21} x \left (275+354 x^2\right ) \sqrt {2+x^2-x^4}-\frac {25}{7} x \left (2+x^2-x^4\right )^{3/2}+\frac {1}{105} \int \frac {9040+10225 x^2}{\sqrt {2+x^2-x^4}} \, dx\\ &=\frac {1}{21} x \left (275+354 x^2\right ) \sqrt {2+x^2-x^4}-\frac {25}{7} x \left (2+x^2-x^4\right )^{3/2}+\frac {2}{105} \int \frac {9040+10225 x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx\\ &=\frac {1}{21} x \left (275+354 x^2\right ) \sqrt {2+x^2-x^4}-\frac {25}{7} x \left (2+x^2-x^4\right )^{3/2}-\frac {158}{7} \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx+\frac {2045}{21} \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx\\ &=\frac {1}{21} x \left (275+354 x^2\right ) \sqrt {2+x^2-x^4}-\frac {25}{7} x \left (2+x^2-x^4\right )^{3/2}+\frac {2045}{21} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {79}{7} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\\ \end {align*}
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Mathematica [C] time = 0.09, size = 102, normalized size = 1.38 \[ \frac {-75 x^9-204 x^7+304 x^5+683 x^3-2949 i \sqrt {-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+2045 i \sqrt {-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+250 x}{21 \sqrt {-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (25 \, x^{4} + 70 \, x^{2} + 49\right )} \sqrt {-x^{4} + x^{2} + 2}, x\right ) \]
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 \[ \int \sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 159, normalized size = 2.15 \[ \frac {25 \sqrt {-x^{4}+x^{2}+2}\, x^{5}}{7}+\frac {93 \sqrt {-x^{4}+x^{2}+2}\, x^{3}}{7}+\frac {125 \sqrt {-x^{4}+x^{2}+2}\, x}{21}+\frac {904 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{21 \sqrt {-x^{4}+x^{2}+2}}-\frac {2045 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )+\EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )\right )}{42 \sqrt {-x^{4}+x^{2}+2}} \]
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 \[ \int \sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
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 \[ \int {\left (5\,x^2+7\right )}^2\,\sqrt {-x^4+x^2+2} \,d x \]
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 \[ \int \sqrt {- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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