Optimal. Leaf size=72 \[ \frac {x \left (35-16 x^2\right )}{306 \sqrt {-x^4+x^2+2}}+\frac {1}{102} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {8}{153} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {25}{238} \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1221, 1178, 1180, 524, 424, 419, 1212, 537} \[ \frac {x \left (35-16 x^2\right )}{306 \sqrt {-x^4+x^2+2}}+\frac {1}{102} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {8}{153} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {25}{238} \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right ) \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 524
Rule 537
Rule 1178
Rule 1180
Rule 1212
Rule 1221
Rubi steps
\begin {align*} \int \frac {1}{\left (7+5 x^2\right ) \left (2+x^2-x^4\right )^{3/2}} \, dx &=-\left (\frac {1}{34} \int \frac {-12+5 x^2}{\left (2+x^2-x^4\right )^{3/2}} \, dx\right )-\frac {25}{34} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+x^2-x^4}} \, dx\\ &=\frac {x \left (35-16 x^2\right )}{306 \sqrt {2+x^2-x^4}}+\frac {1}{612} \int \frac {38+32 x^2}{\sqrt {2+x^2-x^4}} \, dx-\frac {25}{17} \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2} \left (7+5 x^2\right )} \, dx\\ &=\frac {x \left (35-16 x^2\right )}{306 \sqrt {2+x^2-x^4}}-\frac {25}{238} \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {1}{306} \int \frac {38+32 x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx\\ &=\frac {x \left (35-16 x^2\right )}{306 \sqrt {2+x^2-x^4}}-\frac {25}{238} \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {1}{51} \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx+\frac {8}{153} \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx\\ &=\frac {x \left (35-16 x^2\right )}{306 \sqrt {2+x^2-x^4}}+\frac {8}{153} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {1}{102} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {25}{238} \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )\\ \end {align*}
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Mathematica [C] time = 0.22, size = 101, normalized size = 1.40 \[ \frac {\frac {490 x}{\sqrt {-x^4+x^2+2}}-\frac {224 x^3}{\sqrt {-x^4+x^2+2}}-357 i \sqrt {2} F\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+224 i \sqrt {2} E\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+225 i \sqrt {2} \Pi \left (\frac {5}{7};i \sinh ^{-1}(x)|-\frac {1}{2}\right )}{4284} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{4} + x^{2} + 2}}{5 \, x^{10} - 3 \, x^{8} - 29 \, x^{6} - x^{4} + 48 \, x^{2} + 28}, x\right ) \]
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 \[ \int \frac {1}{{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 164, normalized size = 2.28 \[ \frac {4 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{153 \sqrt {-x^{4}+x^{2}+2}}+\frac {\sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{204 \sqrt {-x^{4}+x^{2}+2}}-\frac {25 \sqrt {2}\, \sqrt {-\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {\sqrt {2}\, x}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{238 \sqrt {-x^{4}+x^{2}+2}}+\frac {-\frac {8}{153} x^{3}+\frac {35}{306} x}{\sqrt {-x^{4}+x^{2}+2}} \]
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 \[ \int \frac {1}{{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\left (5\,x^2+7\right )\,{\left (-x^4+x^2+2\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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