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

Optimal. Leaf size=81 \[ \frac{1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{315} x \left (669 x^2+1087\right ) \sqrt{-x^4+x^2+2}+\frac{418}{105} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{4432}{315} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*(1087 + 669*x^2)*Sqrt[2 + x^2 - x^4])/315 + (x*(48 + 35*x^2)*(2 + x^2 - x^4)^
(3/2))/63 + (4432*EllipticE[ArcSin[x/Sqrt[2]], -2])/315 + (418*EllipticF[ArcSin[
x/Sqrt[2]], -2])/105

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Rubi [A]  time = 0.178023, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{1}{63} x \left (35 x^2+48\right ) \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{315} x \left (669 x^2+1087\right ) \sqrt{-x^4+x^2+2}+\frac{418}{105} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{4432}{315} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(1087 + 669*x^2)*Sqrt[2 + x^2 - x^4])/315 + (x*(48 + 35*x^2)*(2 + x^2 - x^4)^
(3/2))/63 + (4432*EllipticE[ArcSin[x/Sqrt[2]], -2])/315 + (418*EllipticF[ArcSin[
x/Sqrt[2]], -2])/105

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Rubi in Sympy [A]  time = 29.2614, size = 76, normalized size = 0.94 \[ \frac{x \left (35 x^{2} + 48\right ) \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{63} + \frac{x \left (669 x^{2} + 1087\right ) \sqrt{- x^{4} + x^{2} + 2}}{315} + \frac{4432 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{315} + \frac{418 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{105} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(35*x**2 + 48)*(-x**4 + x**2 + 2)**(3/2)/63 + x*(669*x**2 + 1087)*sqrt(-x**4 +
x**2 + 2)/315 + 4432*elliptic_e(asin(sqrt(2)*x/2), -2)/315 + 418*elliptic_f(asi
n(sqrt(2)*x/2), -2)/105

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Mathematica [C]  time = 0.101102, size = 107, normalized size = 1.32 \[ \frac{175 x^{11}-110 x^9-1674 x^7-438 x^5+4085 x^3-7275 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+4432 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+3134 x}{315 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(3134*x + 4085*x^3 - 438*x^5 - 1674*x^7 - 110*x^9 + 175*x^11 + (4432*I)*Sqrt[4 +
2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (7275*I)*Sqrt[4 + 2*x^2 - 2*x^4]
*EllipticF[I*ArcSinh[x], -1/2])/(315*Sqrt[2 + x^2 - x^4])

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Maple [B]  time = 0.008, size = 176, normalized size = 2.2 \[ -{\frac{13\,{x}^{5}}{63}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{1259\,{x}^{3}}{315}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{1567\,x}{315}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{2843\,\sqrt{2}}{315}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{2216\,\sqrt{2}}{315}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{5\,{x}^{7}}{9}\sqrt{-{x}^{4}+{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-13/63*x^5*(-x^4+x^2+2)^(1/2)+1259/315*x^3*(-x^4+x^2+2)^(1/2)+1567/315*x*(-x^4+x
^2+2)^(1/2)+2843/315*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*E
llipticF(1/2*2^(1/2)*x,I*2^(1/2))-2216/315*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)))-5/9*x^7*(-x^4+x^2+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (5 \, x^{6} + 2 \, x^{4} - 17 \, x^{2} - 14\right )} \sqrt{-x^{4} + x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(5*x^6 + 2*x^4 - 17*x^2 - 14)*sqrt(-x^4 + x^2 + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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