Optimal. Leaf size=93 \[ -\frac {17 \sqrt {-x^4+x^2+2} x}{175 \left (5 x^2+7\right )}-\frac {1}{75} \sqrt {-x^4+x^2+2} x+\frac {458}{875} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {97}{525} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {1241 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{6125} \]
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Rubi [A] time = 0.32, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {1228, 1095, 419, 1132, 493, 424, 1122, 1180, 1223, 1716, 524, 1212, 537} \[ -\frac {17 \sqrt {-x^4+x^2+2} x}{175 \left (5 x^2+7\right )}-\frac {1}{75} \sqrt {-x^4+x^2+2} x+\frac {458}{875} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {97}{525} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {1241 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{6125} \]
Antiderivative was successfully verified.
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Rule 419
Rule 424
Rule 493
Rule 524
Rule 537
Rule 1095
Rule 1122
Rule 1132
Rule 1180
Rule 1212
Rule 1223
Rule 1228
Rule 1716
Rubi steps
\begin {align*} \int \frac {\left (2+x^2-x^4\right )^{3/2}}{\left (7+5 x^2\right )^2} \, dx &=\int \left (\frac {212}{625 \sqrt {2+x^2-x^4}}-\frac {24 x^2}{125 \sqrt {2+x^2-x^4}}+\frac {x^4}{25 \sqrt {2+x^2-x^4}}+\frac {1156}{625 \left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}}-\frac {1292}{625 \left (7+5 x^2\right ) \sqrt {2+x^2-x^4}}\right ) \, dx\\ &=\frac {1}{25} \int \frac {x^4}{\sqrt {2+x^2-x^4}} \, dx-\frac {24}{125} \int \frac {x^2}{\sqrt {2+x^2-x^4}} \, dx+\frac {212}{625} \int \frac {1}{\sqrt {2+x^2-x^4}} \, dx+\frac {1156}{625} \int \frac {1}{\left (7+5 x^2\right )^2 \sqrt {2+x^2-x^4}} \, dx-\frac {1292}{625} \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+x^2-x^4}} \, dx\\ &=-\frac {1}{75} x \sqrt {2+x^2-x^4}-\frac {17 x \sqrt {2+x^2-x^4}}{175 \left (7+5 x^2\right )}+\frac {17 \int \frac {118-70 x^2-25 x^4}{\left (7+5 x^2\right ) \sqrt {2+x^2-x^4}} \, dx}{4375}+\frac {1}{75} \int \frac {2+2 x^2}{\sqrt {2+x^2-x^4}} \, dx-\frac {48}{125} \int \frac {x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx+\frac {424}{625} \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx-\frac {2584}{625} \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2} \left (7+5 x^2\right )} \, dx\\ &=-\frac {1}{75} x \sqrt {2+x^2-x^4}-\frac {17 x \sqrt {2+x^2-x^4}}{175 \left (7+5 x^2\right )}+\frac {212}{625} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {1292 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{4375}-\frac {17 \int \frac {175+125 x^2}{\sqrt {2+x^2-x^4}} \, dx}{109375}+\frac {2}{75} \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx-\frac {24}{125} \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx+\frac {48}{125} \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx+\frac {2839 \int \frac {1}{\left (7+5 x^2\right ) \sqrt {2+x^2-x^4}} \, dx}{4375}\\ &=-\frac {1}{75} x \sqrt {2+x^2-x^4}-\frac {17 x \sqrt {2+x^2-x^4}}{175 \left (7+5 x^2\right )}-\frac {62}{375} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {332}{625} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {1292 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{4375}-\frac {34 \int \frac {175+125 x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx}{109375}+\frac {5678 \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2} \left (7+5 x^2\right )} \, dx}{4375}\\ &=-\frac {1}{75} x \sqrt {2+x^2-x^4}-\frac {17 x \sqrt {2+x^2-x^4}}{175 \left (7+5 x^2\right )}-\frac {62}{375} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {332}{625} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {1241 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{6125}-\frac {68 \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx}{4375}-\frac {17}{875} \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx\\ &=-\frac {1}{75} x \sqrt {2+x^2-x^4}-\frac {17 x \sqrt {2+x^2-x^4}}{175 \left (7+5 x^2\right )}-\frac {97}{525} E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )+\frac {458}{875} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )-\frac {1241 \Pi \left (-\frac {10}{7};\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{6125}\\ \end {align*}
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Mathematica [C] time = 0.31, size = 201, normalized size = 2.16 \[ \frac {2450 x^7+4550 x^5-11900 x^3+567 i \sqrt {2} \left (5 x^2+7\right ) \sqrt {-x^4+x^2+2} F\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )-6790 i \sqrt {2} \left (5 x^2+7\right ) \sqrt {-x^4+x^2+2} E\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )+18615 i \sqrt {2} \sqrt {-x^4+x^2+2} x^2 \Pi \left (\frac {5}{7};i \sinh ^{-1}(x)|-\frac {1}{2}\right )+26061 i \sqrt {2} \sqrt {-x^4+x^2+2} \Pi \left (\frac {5}{7};i \sinh ^{-1}(x)|-\frac {1}{2}\right )-14000 x}{36750 \left (5 x^2+7\right ) \sqrt {-x^4+x^2+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}}{25 \, x^{4} + 70 \, x^{2} + 49}, x\right ) \]
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 \[ \int \frac {{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 180, normalized size = 1.94 \[ -\frac {17 \sqrt {-x^{4}+x^{2}+2}\, x}{175 \left (5 x^{2}+7\right )}-\frac {\sqrt {-x^{4}+x^{2}+2}\, x}{75}-\frac {97 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticE \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{1050 \sqrt {-x^{4}+x^{2}+2}}+\frac {229 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{875 \sqrt {-x^{4}+x^{2}+2}}-\frac {1241 \sqrt {2}\, \sqrt {-\frac {x^{2}}{2}+1}\, \sqrt {x^{2}+1}\, \EllipticPi \left (\frac {\sqrt {2}\, x}{2}, -\frac {10}{7}, i \sqrt {2}\right )}{6125 \sqrt {-x^{4}+x^{2}+2}} \]
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 \[ \int \frac {{\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (-x^4+x^2+2\right )}^{3/2}}{{\left (5\,x^2+7\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{2}}}{\left (5 x^{2} + 7\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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