Optimal. Leaf size=100 \[ -\frac {25}{11} x \left (-x^4+x^2+2\right )^{5/2}+\frac {1}{99} x \left (920 x^2+363\right ) \left (-x^4+x^2+2\right )^{3/2}+\frac {1}{495} x \left (14889 x^2+11497\right ) \sqrt {-x^4+x^2+2}-\frac {3392}{165} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {85942}{495} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 7, 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} \[ -\frac {25}{11} x \left (-x^4+x^2+2\right )^{5/2}+\frac {1}{99} x \left (920 x^2+363\right ) \left (-x^4+x^2+2\right )^{3/2}+\frac {1}{495} x \left (14889 x^2+11497\right ) \sqrt {-x^4+x^2+2}-\frac {3392}{165} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {85942}{495} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
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 \left (2+x^2-x^4\right )^{3/2} \, dx &=-\frac {25}{11} x \left (2+x^2-x^4\right )^{5/2}-\frac {1}{11} \int \left (-589-920 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx\\ &=\frac {1}{99} x \left (363+920 x^2\right ) \left (2+x^2-x^4\right )^{3/2}-\frac {25}{11} x \left (2+x^2-x^4\right )^{5/2}+\frac {1}{231} \int \left (23044+34741 x^2\right ) \sqrt {2+x^2-x^4} \, dx\\ &=\frac {1}{495} x \left (11497+14889 x^2\right ) \sqrt {2+x^2-x^4}+\frac {1}{99} x \left (363+920 x^2\right ) \left (2+x^2-x^4\right )^{3/2}-\frac {25}{11} x \left (2+x^2-x^4\right )^{5/2}-\frac {\int \frac {-530362-601594 x^2}{\sqrt {2+x^2-x^4}} \, dx}{3465}\\ &=\frac {1}{495} x \left (11497+14889 x^2\right ) \sqrt {2+x^2-x^4}+\frac {1}{99} x \left (363+920 x^2\right ) \left (2+x^2-x^4\right )^{3/2}-\frac {25}{11} x \left (2+x^2-x^4\right )^{5/2}-\frac {2 \int \frac {-530362-601594 x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx}{3465}\\ &=\frac {1}{495} x \left (11497+14889 x^2\right ) \sqrt {2+x^2-x^4}+\frac {1}{99} x \left (363+920 x^2\right ) \left (2+x^2-x^4\right )^{3/2}-\frac {25}{11} x \left (2+x^2-x^4\right )^{5/2}-\frac {6784}{165} \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx+\frac {85942}{495} \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx\\ &=\frac {1}{495} x \left (11497+14889 x^2\right ) \sqrt {2+x^2-x^4}+\frac {1}{99} x \left (363+920 x^2\right ) \left (2+x^2-x^4\right )^{3/2}-\frac {25}{11} x \left (2+x^2-x^4\right )^{5/2}+\frac {85942}{495} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {3392}{165} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\\ \end {align*}
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Mathematica [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (25 \, x^{8} + 45 \, x^{6} - 71 \, x^{4} - 189 \, x^{2} - 98\right )} \sqrt {-x^{4} + x^{2} + 2}, x\right ) \]
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 \[ \int {\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.01, size = 193, normalized size = 1.93 \[ -\frac {25 \sqrt {-x^{4}+x^{2}+2}\, x^{9}}{11}-\frac {470 \sqrt {-x^{4}+x^{2}+2}\, x^{7}}{99}+\frac {112 \sqrt {-x^{4}+x^{2}+2}\, x^{5}}{9}+\frac {21404 \sqrt {-x^{4}+x^{2}+2}\, x^{3}}{495}+\frac {10627 \sqrt {-x^{4}+x^{2}+2}\, x}{495}+\frac {37883 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{495 \sqrt {-x^{4}+x^{2}+2}}-\frac {42971 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )+\EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )\right )}{495 \sqrt {-x^{4}+x^{2}+2}} \]
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 \[ \int {\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{2}\,{d x} \]
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 \[ \int {\left (5\,x^2+7\right )}^2\,{\left (-x^4+x^2+2\right )}^{3/2} \,d x \]
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 \[ \int \left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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