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

Optimal. Leaf size=100 \[ -\frac{25}{11} x \left (-x^4+x^2+2\right )^{5/2}+\frac{1}{99} x \left (920 x^2+363\right ) \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{495} x \left (14889 x^2+11497\right ) \sqrt{-x^4+x^2+2}-\frac{3392}{165} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{85942}{495} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*(11497 + 14889*x^2)*Sqrt[2 + x^2 - x^4])/495 + (x*(363 + 920*x^2)*(2 + x^2 -
x^4)^(3/2))/99 - (25*x*(2 + x^2 - x^4)^(5/2))/11 + (85942*EllipticE[ArcSin[x/Sqr
t[2]], -2])/495 - (3392*EllipticF[ArcSin[x/Sqrt[2]], -2])/165

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Rubi [A]  time = 0.220222, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{25}{11} x \left (-x^4+x^2+2\right )^{5/2}+\frac{1}{99} x \left (920 x^2+363\right ) \left (-x^4+x^2+2\right )^{3/2}+\frac{1}{495} x \left (14889 x^2+11497\right ) \sqrt{-x^4+x^2+2}-\frac{3392}{165} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{85942}{495} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(11497 + 14889*x^2)*Sqrt[2 + x^2 - x^4])/495 + (x*(363 + 920*x^2)*(2 + x^2 -
x^4)^(3/2))/99 - (25*x*(2 + x^2 - x^4)^(5/2))/11 + (85942*EllipticE[ArcSin[x/Sqr
t[2]], -2])/495 - (3392*EllipticF[ArcSin[x/Sqrt[2]], -2])/165

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Rubi in Sympy [A]  time = 38.9803, size = 99, normalized size = 0.99 \[ \frac{x \left (\frac{6440 x^{2}}{11} + 231\right ) \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{63} + \frac{x \left (\frac{104223 x^{2}}{11} + \frac{80479}{11}\right ) \sqrt{- x^{4} + x^{2} + 2}}{315} - \frac{25 x \left (- x^{4} + x^{2} + 2\right )^{\frac{5}{2}}}{11} + \frac{85942 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{495} - \frac{3392 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{165} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(6440*x**2/11 + 231)*(-x**4 + x**2 + 2)**(3/2)/63 + x*(104223*x**2/11 + 80479/
11)*sqrt(-x**4 + x**2 + 2)/315 - 25*x*(-x**4 + x**2 + 2)**(5/2)/11 + 85942*ellip
tic_e(asin(sqrt(2)*x/2), -2)/495 - 3392*elliptic_f(asin(sqrt(2)*x/2), -2)/165

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Mathematica [C]  time = 0.11192, size = 112, normalized size = 1.12 \[ \frac{1125 x^{13}+1225 x^{11}-10760 x^9-19944 x^7+23097 x^5+53435 x^3-123825 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+85942 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+21254 x}{495 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(21254*x + 53435*x^3 + 23097*x^5 - 19944*x^7 - 10760*x^9 + 1225*x^11 + 1125*x^13
+ (85942*I)*Sqrt[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (123825*I)*
Sqrt[4 + 2*x^2 - 2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(495*Sqrt[2 + x^2 - x^4])

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Maple [B]  time = 0.011, size = 193, normalized size = 1.9 \[{\frac{112\,{x}^{5}}{9}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{21404\,{x}^{3}}{495}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{10627\,x}{495}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{37883\,\sqrt{2}}{495}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{42971\,\sqrt{2}}{495}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{470\,{x}^{7}}{99}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{25\,{x}^{9}}{11}\sqrt{-{x}^{4}+{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

112/9*x^5*(-x^4+x^2+2)^(1/2)+21404/495*x^3*(-x^4+x^2+2)^(1/2)+10627/495*x*(-x^4+
x^2+2)^(1/2)+37883/495*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))-42971/495*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)))-470/99*x^7*(-x^4+x^2+2)^(1/2)-25/11*x^9*(-x^4+x^2+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (25 \, x^{8} + 45 \, x^{6} - 71 \, x^{4} - 189 \, x^{2} - 98\right )} \sqrt{-x^{4} + x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(25*x^8 + 45*x^6 - 71*x^4 - 189*x^2 - 98)*sqrt(-x^4 + x^2 + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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