Optimal. Leaf size=65 \[ -\frac {625}{3} \sqrt {-x^4+x^2+2} x-25 \sqrt {-x^4+x^2+2} x^3-542 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {3905}{3} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.08, antiderivative size = 65, 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, 1679, 1180, 524, 424, 419} \[ -25 \sqrt {-x^4+x^2+2} x^3-\frac {625}{3} \sqrt {-x^4+x^2+2} x-542 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {3905}{3} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 1180
Rule 1206
Rule 1679
Rubi steps
\begin {align*} \int \frac {\left (7+5 x^2\right )^3}{\sqrt {2+x^2-x^4}} \, dx &=-25 x^3 \sqrt {2+x^2-x^4}-\frac {1}{5} \int \frac {-1715-4425 x^2-3125 x^4}{\sqrt {2+x^2-x^4}} \, dx\\ &=-\frac {625}{3} x \sqrt {2+x^2-x^4}-25 x^3 \sqrt {2+x^2-x^4}+\frac {1}{15} \int \frac {11395+19525 x^2}{\sqrt {2+x^2-x^4}} \, dx\\ &=-\frac {625}{3} x \sqrt {2+x^2-x^4}-25 x^3 \sqrt {2+x^2-x^4}+\frac {2}{15} \int \frac {11395+19525 x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx\\ &=-\frac {625}{3} x \sqrt {2+x^2-x^4}-25 x^3 \sqrt {2+x^2-x^4}-1084 \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx+\frac {3905}{3} \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx\\ &=-\frac {625}{3} x \sqrt {2+x^2-x^4}-25 x^3 \sqrt {2+x^2-x^4}+\frac {3905}{3} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-542 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\\ \end {align*}
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Mathematica [C] time = 0.11, size = 97, normalized size = 1.49 \[ \frac {150 x^7+1100 x^5-1550 x^3-10089 i \sqrt {-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+7810 i \sqrt {-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )-2500 x}{6 \sqrt {-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343\right )} \sqrt {-x^{4} + x^{2} + 2}}{x^{4} - x^{2} - 2}, x\right ) \]
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 \[ \int \frac {{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt {-x^{4} + x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 142, normalized size = 2.18 \[ -25 \sqrt {-x^{4}+x^{2}+2}\, x^{3}-\frac {625 \sqrt {-x^{4}+x^{2}+2}\, x}{3}+\frac {2279 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{6 \sqrt {-x^{4}+x^{2}+2}}-\frac {3905 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )+\EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )\right )}{6 \sqrt {-x^{4}+x^{2}+2}} \]
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 \[ \int \frac {{\left (5 \, x^{2} + 7\right )}^{3}}{\sqrt {-x^{4} + x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (5\,x^2+7\right )}^3}{\sqrt {-x^4+x^2+2}} \,d x \]
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 \[ \int \frac {\left (5 x^{2} + 7\right )^{3}}{\sqrt {- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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