Optimal. Leaf size=142 \[ -\frac {132300}{143} \left (-x^4+x^2+2\right )^{5/2} x-\frac {\left (69817-1581440 x^2\right ) \left (-x^4+x^2+2\right )^{3/2} x}{1001}+\frac {3 \left (7837383 x^2+2193559\right ) \sqrt {-x^4+x^2+2} x}{5005}-\frac {125}{3} \left (-x^4+x^2+2\right )^{5/2} x^5-\frac {11750}{39} \left (-x^4+x^2+2\right )^{5/2} x^3-\frac {50794416 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{5005}+\frac {124141422 E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{5005} \]
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Rubi [A] time = 0.13, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ -\frac {125}{3} \left (-x^4+x^2+2\right )^{5/2} x^5-\frac {11750}{39} \left (-x^4+x^2+2\right )^{5/2} x^3-\frac {132300}{143} \left (-x^4+x^2+2\right )^{5/2} x-\frac {\left (69817-1581440 x^2\right ) \left (-x^4+x^2+2\right )^{3/2} x}{1001}+\frac {3 \left (7837383 x^2+2193559\right ) \sqrt {-x^4+x^2+2} x}{5005}-\frac {50794416 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{5005}+\frac {124141422 E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{5005} \]
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 )^4 \left (2+x^2-x^4\right )^{3/2} \, dx &=-\frac {125}{3} x^5 \left (2+x^2-x^4\right )^{5/2}-\frac {1}{15} \int \left (2+x^2-x^4\right )^{3/2} \left (-36015-102900 x^2-116500 x^4-58750 x^6\right ) \, dx\\ &=-\frac {11750}{39} x^3 \left (2+x^2-x^4\right )^{5/2}-\frac {125}{3} x^5 \left (2+x^2-x^4\right )^{5/2}+\frac {1}{195} \int \left (2+x^2-x^4\right )^{3/2} \left (468195+1690200 x^2+1984500 x^4\right ) \, dx\\ &=-\frac {132300}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac {11750}{39} x^3 \left (2+x^2-x^4\right )^{5/2}-\frac {125}{3} x^5 \left (2+x^2-x^4\right )^{5/2}-\frac {\int \left (-9119145-30499200 x^2\right ) \left (2+x^2-x^4\right )^{3/2} \, dx}{2145}\\ &=-\frac {x \left (69817-1581440 x^2\right ) \left (2+x^2-x^4\right )^{3/2}}{1001}-\frac {132300}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac {11750}{39} x^3 \left (2+x^2-x^4\right )^{5/2}-\frac {125}{3} x^5 \left (2+x^2-x^4\right )^{5/2}+\frac {\int \left (389287620+1058046705 x^2\right ) \sqrt {2+x^2-x^4} \, dx}{45045}\\ &=\frac {3 x \left (2193559+7837383 x^2\right ) \sqrt {2+x^2-x^4}}{5005}-\frac {x \left (69817-1581440 x^2\right ) \left (2+x^2-x^4\right )^{3/2}}{1001}-\frac {132300}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac {11750}{39} x^3 \left (2+x^2-x^4\right )^{5/2}-\frac {125}{3} x^5 \left (2+x^2-x^4\right )^{5/2}-\frac {\int \frac {-9901845810-16759091970 x^2}{\sqrt {2+x^2-x^4}} \, dx}{675675}\\ &=\frac {3 x \left (2193559+7837383 x^2\right ) \sqrt {2+x^2-x^4}}{5005}-\frac {x \left (69817-1581440 x^2\right ) \left (2+x^2-x^4\right )^{3/2}}{1001}-\frac {132300}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac {11750}{39} x^3 \left (2+x^2-x^4\right )^{5/2}-\frac {125}{3} x^5 \left (2+x^2-x^4\right )^{5/2}-\frac {2 \int \frac {-9901845810-16759091970 x^2}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx}{675675}\\ &=\frac {3 x \left (2193559+7837383 x^2\right ) \sqrt {2+x^2-x^4}}{5005}-\frac {x \left (69817-1581440 x^2\right ) \left (2+x^2-x^4\right )^{3/2}}{1001}-\frac {132300}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac {11750}{39} x^3 \left (2+x^2-x^4\right )^{5/2}-\frac {125}{3} x^5 \left (2+x^2-x^4\right )^{5/2}-\frac {101588832 \int \frac {1}{\sqrt {4-2 x^2} \sqrt {2+2 x^2}} \, dx}{5005}+\frac {124141422 \int \frac {\sqrt {2+2 x^2}}{\sqrt {4-2 x^2}} \, dx}{5005}\\ &=\frac {3 x \left (2193559+7837383 x^2\right ) \sqrt {2+x^2-x^4}}{5005}-\frac {x \left (69817-1581440 x^2\right ) \left (2+x^2-x^4\right )^{3/2}}{1001}-\frac {132300}{143} x \left (2+x^2-x^4\right )^{5/2}-\frac {11750}{39} x^3 \left (2+x^2-x^4\right )^{5/2}-\frac {125}{3} x^5 \left (2+x^2-x^4\right )^{5/2}+\frac {124141422 E\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{5005}-\frac {50794416 F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{5005}\\ \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 = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (625 \, x^{12} + 2875 \, x^{10} + 2600 \, x^{8} - 7490 \, x^{6} - 19159 \, x^{4} - 16121 \, x^{2} - 4802\right )} \sqrt {-x^{4} + x^{2} + 2}, x\right ) \]
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 \[ \int {\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 227, normalized size = 1.60 \[ -\frac {125 \sqrt {-x^{4}+x^{2}+2}\, x^{13}}{3}-\frac {8500 \sqrt {-x^{4}+x^{2}+2}\, x^{11}}{39}-\frac {84775 \sqrt {-x^{4}+x^{2}+2}\, x^{9}}{429}+\frac {432290 \sqrt {-x^{4}+x^{2}+2}\, x^{7}}{429}+\frac {833561 \sqrt {-x^{4}+x^{2}+2}\, x^{5}}{273}+\frac {43271392 \sqrt {-x^{4}+x^{2}+2}\, x^{3}}{15015}-\frac {12639493 \sqrt {-x^{4}+x^{2}+2}\, x}{5005}+\frac {36673503 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )}{5005 \sqrt {-x^{4}+x^{2}+2}}-\frac {62070711 \sqrt {2}\, \sqrt {-2 x^{2}+4}\, \sqrt {x^{2}+1}\, \left (-\EllipticE \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )+\EllipticF \left (\frac {\sqrt {2}\, x}{2}, i \sqrt {2}\right )\right )}{5005 \sqrt {-x^{4}+x^{2}+2}} \]
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 \[ \int {\left (-x^{4} + x^{2} + 2\right )}^{\frac {3}{2}} {\left (5 \, x^{2} + 7\right )}^{4}\,{d x} \]
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 \[ \int {\left (5\,x^2+7\right )}^4\,{\left (-x^4+x^2+2\right )}^{3/2} \,d x \]
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 \[ \int \left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac {3}{2}} \left (5 x^{2} + 7\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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