3.328 \(\int \left (7+5 x^2\right )^4 \left (2+x^2-x^4\right )^{3/2} \, dx\)

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]

(3*x*(2193559 + 7837383*x^2)*Sqrt[2 + x^2 - x^4])/5005 - (x*(69817 - 1581440*x^2
)*(2 + x^2 - x^4)^(3/2))/1001 - (132300*x*(2 + x^2 - x^4)^(5/2))/143 - (11750*x^
3*(2 + x^2 - x^4)^(5/2))/39 - (125*x^5*(2 + x^2 - x^4)^(5/2))/3 + (124141422*Ell
ipticE[ArcSin[x/Sqrt[2]], -2])/5005 - (50794416*EllipticF[ArcSin[x/Sqrt[2]], -2]
)/5005

_______________________________________________________________________________________

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]

(3*x*(2193559 + 7837383*x^2)*Sqrt[2 + x^2 - x^4])/5005 - (x*(69817 - 1581440*x^2
)*(2 + x^2 - x^4)^(3/2))/1001 - (132300*x*(2 + x^2 - x^4)^(5/2))/143 - (11750*x^
3*(2 + x^2 - x^4)^(5/2))/39 - (125*x^5*(2 + x^2 - x^4)^(5/2))/3 + (124141422*Ell
ipticE[ArcSin[x/Sqrt[2]], -2])/5005 - (50794416*EllipticF[ArcSin[x/Sqrt[2]], -2]
)/5005

_______________________________________________________________________________________

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]

-125*x**5*(-x**4 + x**2 + 2)**(5/2)/3 - 11750*x**3*(-x**4 + x**2 + 2)**(5/2)/39
- x*(-14232960*x**2/143 + 57123/13)*(-x**4 + x**2 + 2)**(3/2)/63 + x*(211609341*
x**2/143 + 59226093/143)*sqrt(-x**4 + x**2 + 2)/315 - 132300*x*(-x**4 + x**2 + 2
)**(5/2)/143 + 124141422*elliptic_e(asin(sqrt(2)*x/2), -2)/5005 - 50794416*ellip
tic_f(asin(sqrt(2)*x/2), -2)/5005

_______________________________________________________________________________________

Mathematica [C]  time = 0.117841, size = 122, normalized size = 0.86 \[ \frac{625625 x^{17}+2646875 x^{15}-1556625 x^{13}-24642275 x^{11}-36649955 x^9+32834763 x^7+172881581 x^5+48624305 x^3-482444775 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )[phone] i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-75836958 x}{15015 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(7 + 5*x^2)^4*(2 + x^2 - x^4)^(3/2),x]

[Out]

(-75836958*x + 48624305*x^3 + 172881581*x^5 + 32834763*x^7 - 36649955*x^9 - 2464
2275*x^11 - 1556625*x^13 + 2646875*x^15 + 625625*x^17 + (372424266*I)*Sqrt[4 + 2
*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (482444775*I)*Sqrt[4 + 2*x^2 - 2*x
^4]*EllipticF[I*ArcSinh[x], -1/2])/(15015*Sqrt[2 + x^2 - x^4])

_______________________________________________________________________________________

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]

833561/273*x^5*(-x^4+x^2+2)^(1/2)[phone]/15015*x^3*(-x^4+x^2+2)^(1/2)-12639493
/5005*x*(-x^4+x^2+2)^(1/2)[phone]/5005*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/
(-x^4+x^2+2)^(1/2)*EllipticF(1/2*2^(1/2)*x,I*2^(1/2))-62070711/5005*2^(1/2)*(-2*
x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*(EllipticF(1/2*2^(1/2)*x,I*2^(1/2)
)-EllipticE(1/2*2^(1/2)*x,I*2^(1/2)))+432290/429*x^7*(-x^4+x^2+2)^(1/2)-84775/42
9*x^9*(-x^4+x^2+2)^(1/2)-8500/39*x^11*(-x^4+x^2+2)^(1/2)-125/3*x^13*(-x^4+x^2+2)
^(1/2)

_______________________________________________________________________________________

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]

integrate((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)^4, x)

_______________________________________________________________________________________

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]

integral(-(625*x^12 + 2875*x^10 + 2600*x^8 - 7490*x^6 - 19159*x^4 - 16121*x^2 -
4802)*sqrt(-x^4 + x^2 + 2), x)

_______________________________________________________________________________________

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]

Integral((-(x**2 - 2)*(x**2 + 1))**(3/2)*(5*x**2 + 7)**4, x)

_______________________________________________________________________________________

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]

integrate((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)^4, x)