Optimal. Leaf size=95 \[ -\frac {1825}{21} \left (-x^4+x^2+2\right )^{3/2} x+\frac {1}{63} \left (14691 x^2+5956\right ) \sqrt {-x^4+x^2+2} x-\frac {125}{9} \left (-x^4+x^2+2\right )^{3/2} x^3-\frac {8735}{21} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {79411}{63} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {1206, 1679, 1176, 1180, 524, 424, 419} \[ -\frac {125}{9} \left (-x^4+x^2+2\right )^{3/2} x^3-\frac {1825}{21} \left (-x^4+x^2+2\right )^{3/2} x+\frac {1}{63} \left (14691 x^2+5956\right ) \sqrt {-x^4+x^2+2} x-\frac {8735}{21} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {79411}{63} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 1176
Rule 1180
Rule 1206
Rule 1679
Rubi steps
\begin {align*} \int \left (7+5 x^2\right )^3 \sqrt {2+x^2-x^4} \, dx &=-\frac {125}{9} x^3 \left (2+x^2-x^4\right )^{3/2}-\frac {1}{9} \int \left (-3087-7365 x^2-5475 x^4\right ) \sqrt {2+x^2-x^4} \, dx\\ &=-\frac {1825}{21} x \left (2+x^2-x^4\right )^{3/2}-\frac {125}{9} x^3 \left (2+x^2-x^4\right )^{3/2}+\frac {1}{63} \int \left (32559+73455 x^2\right ) \sqrt {2+x^2-x^4} \, dx\\ &=\frac {1}{63} x \left (5956+14691 x^2\right ) \sqrt {2+x^2-x^4}-\frac {1825}{21} x \left (2+x^2-x^4\right )^{3/2}-\frac {125}{9} x^3 \left (2+x^2-x^4\right )^{3/2}-\frac {1}{945} \int \frac {-798090-1191165 x^2}{\sqrt {2+x^2-x^4}} \, dx\\ &=\frac {1}{63} x \left (5956+14691 x^2\right ) \sqrt {2+x^2-x^4}-\frac {1825}{21} x \left (2+x^2-x^4\right )^{3/2}-\frac {125}{9} x^3 \left (2+x^2-x^4\right )^{3/2}-\frac {2}{945} \int \frac {-798090-1191165 x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx\\ &=\frac {1}{63} x \left (5956+14691 x^2\right ) \sqrt {2+x^2-x^4}-\frac {1825}{21} x \left (2+x^2-x^4\right )^{3/2}-\frac {125}{9} x^3 \left (2+x^2-x^4\right )^{3/2}-\frac {17470}{21} \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx+\frac {79411}{63} \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx\\ &=\frac {1}{63} x \left (5956+14691 x^2\right ) \sqrt {2+x^2-x^4}-\frac {1825}{21} x \left (2+x^2-x^4\right )^{3/2}-\frac {125}{9} x^3 \left (2+x^2-x^4\right )^{3/2}+\frac {79411}{63} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {8735}{21} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\\ \end {align*}
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Mathematica [C] time = 0.10, size = 107, normalized size = 1.13 \[ \frac {-875 x^{11}-3725 x^9-1116 x^7+21660 x^5+9938 x^3-106014 i \sqrt {-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+79411 i \sqrt {-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )-9988 x}{63 \sqrt {-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343\right )} \sqrt {-x^{4} + x^{2} + 2}, x\right ) \]
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 \[ \int \sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 176, normalized size = 1.85 \[ \frac {125 \sqrt {-x^{4}+x^{2}+2}\, x^{7}}{9}+\frac {4600 \sqrt {-x^{4}+x^{2}+2}\, x^{5}}{63}+\frac {7466 \sqrt {-x^{4}+x^{2}+2}\, x^{3}}{63}-\frac {4994 \sqrt {-x^{4}+x^{2}+2}\, x}{63}+\frac {26603 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{63 \sqrt {-x^{4}+x^{2}+2}}-\frac {79411 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )+\EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )\right )}{126 \sqrt {-x^{4}+x^{2}+2}} \]
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 \[ \int \sqrt {-x^{4} + x^{2} + 2} {\left (5 \, x^{2} + 7\right )}^{3}\,{d x} \]
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 \[ \int {\left (5\,x^2+7\right )}^3\,\sqrt {-x^4+x^2+2} \,d x \]
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 \[ \int \sqrt {- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )} \left (5 x^{2} + 7\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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