Optimal. Leaf size=142 \[ -\frac{50794416 \text{EllipticF}\left (\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right ),-2\right )}{5005}-\frac{125}{3} \left (-x^4+x^2+2\right )^{5/2} x^5-\frac{11750}{39} \left (-x^4+x^2+2\right )^{5/2} x^3-\frac{132300}{143} \left (-x^4+x^2+2\right )^{5/2} x-\frac{\left (69817-1581440 x^2\right ) \left (-x^4+x^2+2\right )^{3/2} x}{1001}+\frac{3 \left (7837383 x^2+2193559\right ) \sqrt{-x^4+x^2+2} x}{5005}+\frac{124141422 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{5005} \]
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Rubi [A] time = 0.131612, antiderivative size = 142, normalized size of antiderivative = 1., 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 1206
Rule 1679
Rule 1176
Rule 1180
Rule 524
Rule 424
Rule 419
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*}
Mathematica [F] time = 0, size = 0, normalized size = 0. \[ \text{\$Aborted} \]
Verification is Not applicable to the result.
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Maple [A] time = 0.023, size = 227, normalized size = 1.6 \begin{align*} -{\frac{12639493\,x}{5005}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{125\,{x}^{13}}{3}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{833561\,{x}^{5}}{273}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{8500\,{x}^{11}}{39}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{43271392\,{x}^{3}}{15015}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{84775\,{x}^{9}}{429}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{432290\,{x}^{7}}{429}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{62070711\,\sqrt{2}}{5005}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{36673503\,\sqrt{2}}{5005}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{x\sqrt{2}}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )^{4}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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