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

Optimal. Leaf size=121 \[ -\frac{7825}{143} \left (-x^4+x^2+2\right )^{5/2} x+\frac{\left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2} x}{3003}+\frac{\left (5712051 x^2+2512273\right ) \sqrt{-x^4+x^2+2} x}{15015}-\frac{125}{13} \left (-x^4+x^2+2\right )^{5/2} x^3-\frac{3199778 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{5005}+\frac{31072528 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{15015} \]

[Out]

(x*(2512273 + 5712051*x^2)*Sqrt[2 + x^2 - x^4])/15015 + (x*(33792 + 374045*x^2)*
(2 + x^2 - x^4)^(3/2))/3003 - (7825*x*(2 + x^2 - x^4)^(5/2))/143 - (125*x^3*(2 +
x^2 - x^4)^(5/2))/13 + (31072528*EllipticE[ArcSin[x/Sqrt[2]], -2])/15015 - (319
9778*EllipticF[ArcSin[x/Sqrt[2]], -2])/5005

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Rubi [A]  time = 0.253102, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{7825}{143} \left (-x^4+x^2+2\right )^{5/2} x+\frac{\left (374045 x^2+33792\right ) \left (-x^4+x^2+2\right )^{3/2} x}{3003}+\frac{\left (5712051 x^2+2512273\right ) \sqrt{-x^4+x^2+2} x}{15015}-\frac{125}{13} \left (-x^4+x^2+2\right )^{5/2} x^3-\frac{3199778 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{5005}+\frac{31072528 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{15015} \]

Antiderivative was successfully verified.

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

[Out]

(x*(2512273 + 5712051*x^2)*Sqrt[2 + x^2 - x^4])/15015 + (x*(33792 + 374045*x^2)*
(2 + x^2 - x^4)^(3/2))/3003 - (7825*x*(2 + x^2 - x^4)^(5/2))/143 - (125*x^3*(2 +
x^2 - x^4)^(5/2))/13 + (31072528*EllipticE[ArcSin[x/Sqrt[2]], -2])/15015 - (319
9778*EllipticF[ArcSin[x/Sqrt[2]], -2])/5005

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Rubi in Sympy [A]  time = 48.2813, size = 119, normalized size = 0.98 \[ - \frac{125 x^{3} \left (- x^{4} + x^{2} + 2\right )^{\frac{5}{2}}}{13} + \frac{x \left (\frac{1122135 x^{2}}{143} + \frac{9216}{13}\right ) \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{63} + \frac{x \left (\frac{17136153 x^{2}}{143} + \frac{7536819}{143}\right ) \sqrt{- x^{4} + x^{2} + 2}}{315} - \frac{7825 x \left (- x^{4} + x^{2} + 2\right )^{\frac{5}{2}}}{143} + \frac{31072528 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{15015} - \frac{3199778 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{5005} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-125*x**3*(-x**4 + x**2 + 2)**(5/2)/13 + x*(1122135*x**2/143 + 9216/13)*(-x**4 +
x**2 + 2)**(3/2)/63 + x*(17136153*x**2/143 + 7536819/143)*sqrt(-x**4 + x**2 + 2
)/315 - 7825*x*(-x**4 + x**2 + 2)**(5/2)/143 + 31072528*elliptic_e(asin(sqrt(2)*
x/2), -2)/15015 - 3199778*elliptic_f(asin(sqrt(2)*x/2), -2)/5005

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Mathematica [C]  time = 0.11404, size = 117, normalized size = 0.97 \[ \frac{144375 x^{15}+388500 x^{13}-1027775 x^{11}-4448240 x^9-1756521 x^7+13371048 x^5+11078615 x^3-41809125 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )[phone] i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-872614 x}{15015 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-872614*x + 11078615*x^3 + 13371048*x^5 - 1756521*x^7 - 4448240*x^9 - 1027775*x
^11 + 388500*x^13 + 144375*x^15 + (31072528*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE
[I*ArcSinh[x], -1/2] - (41809125*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[
x], -1/2])/(15015*Sqrt[2 + x^2 - x^4])

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Maple [A]  time = 0.011, size = 210, normalized size = 1.7 \[{\frac{65248\,{x}^{5}}{273}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{5757461\,{x}^{3}}{15015}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{436307\,x}{15015}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{10736597\,\sqrt{2}}{15015}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{15536264\,\sqrt{2}}{15015}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{5890\,{x}^{7}}{429}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{5075\,{x}^{9}}{143}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{125\,{x}^{11}}{13}\sqrt{-{x}^{4}+{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

65248/273*x^5*(-x^4+x^2+2)^(1/2)+5757461/15015*x^3*(-x^4+x^2+2)^(1/2)-436307/150
15*x*(-x^4+x^2+2)^(1/2)[phone]/15015*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))-15536264/15015*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)))+5890/429*x^7*(-x^4+x^2+2)^(1/2)-5075/143*x^
9*(-x^4+x^2+2)^(1/2)-125/13*x^11*(-x^4+x^2+2)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)^3,x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (125 \, x^{10} + 400 \, x^{8} - 40 \, x^{6} - 1442 \, x^{4} - 1813 \, x^{2} - 686\right )} \sqrt{-x^{4} + x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)^3,x, algorithm="fricas")

[Out]

integral(-(125*x^10 + 400*x^8 - 40*x^6 - 1442*x^4 - 1813*x^2 - 686)*sqrt(-x^4 +
x^2 + 2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)^3,x, algorithm="giac")

[Out]

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