Optimal. Leaf size=142 \[ -\frac{132300}{143} \left (-x^4+x^2+2\right )^{5/2} x-\frac{\left (69817-1581440 x^2\right ) \left (-x^4+x^2+2\right )^{3/2} x}{1001}+\frac{3 \left (7837383 x^2+2193559\right ) \sqrt{-x^4+x^2+2} x}{5005}-\frac{125}{3} \left (-x^4+x^2+2\right )^{5/2} x^5-\frac{11750}{39} \left (-x^4+x^2+2\right )^{5/2} x^3-\frac{50794416 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{5005}+\frac{124141422 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{5005} \]
[Out]
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Rubi [A] time = 0.288905, 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 \[ -\frac{132300}{143} \left (-x^4+x^2+2\right )^{5/2} x-\frac{\left (69817-1581440 x^2\right ) \left (-x^4+x^2+2\right )^{3/2} x}{1001}+\frac{3 \left (7837383 x^2+2193559\right ) \sqrt{-x^4+x^2+2} x}{5005}-\frac{125}{3} \left (-x^4+x^2+2\right )^{5/2} x^5-\frac{11750}{39} \left (-x^4+x^2+2\right )^{5/2} x^3-\frac{50794416 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{5005}+\frac{124141422 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{5005} \]
Antiderivative was successfully verified.
[In] Int[(7 + 5*x^2)^4*(2 + x^2 - x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 60.5915, size = 138, normalized size = 0.97 \[ - \frac{125 x^{5} \left (- x^{4} + x^{2} + 2\right )^{\frac{5}{2}}}{3} - \frac{11750 x^{3} \left (- x^{4} + x^{2} + 2\right )^{\frac{5}{2}}}{39} - \frac{x \left (- \frac{14232960 x^{2}}{143} + \frac{57123}{13}\right ) \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{63} + \frac{x \left (\frac{211609341 x^{2}}{143} + \frac{59226093}{143}\right ) \sqrt{- x^{4} + x^{2} + 2}}{315} - \frac{132300 x \left (- x^{4} + x^{2} + 2\right )^{\frac{5}{2}}}{143} + \frac{124141422 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{5005} - \frac{50794416 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{5005} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((5*x**2+7)**4*(-x**4+x**2+2)**(3/2),x)
[Out]
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Mathematica [C] time = 0.117841, size = 122, normalized size = 0.86 \[ \frac{625625 x^{17}+2646875 x^{15}-1556625 x^{13}-24642275 x^{11}-36649955 x^9+32834763 x^7+172881581 x^5+48624305 x^3-482444775 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+372424266 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-75836958 x}{15015 \sqrt{-x^4+x^2+2}} \]
Antiderivative was successfully verified.
[In] Integrate[(7 + 5*x^2)^4*(2 + x^2 - x^4)^(3/2),x]
[Out]
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Maple [A] time = 0.032, size = 227, normalized size = 1.6 \[{\frac{833561\,{x}^{5}}{273}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{43271392\,{x}^{3}}{15015}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{12639493\,x}{5005}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{36673503\,\sqrt{2}}{5005}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{62070711\,\sqrt{2}}{5005}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{432290\,{x}^{7}}{429}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{84775\,{x}^{9}}{429}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{8500\,{x}^{11}}{39}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{125\,{x}^{13}}{3}\sqrt{-{x}^{4}+{x}^{2}+2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((5*x^2+7)^4*(-x^4+x^2+2)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)^4,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (625 \, x^{12} + 2875 \, x^{10} + 2600 \, x^{8} - 7490 \, x^{6} - 19159 \, x^{4} - 16121 \, x^{2} - 4802\right )} \sqrt{-x^{4} + x^{2} + 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)^4,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )^{4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x**2+7)**4*(-x**4+x**2+2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)^4,x, algorithm="giac")
[Out]