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

Optimal. Leaf size=95 \[ -\frac{1825}{21} \left (-x^4+x^2+2\right )^{3/2} x+\frac{1}{63} \left (14691 x^2+5956\right ) \sqrt{-x^4+x^2+2} x-\frac{125}{9} \left (-x^4+x^2+2\right )^{3/2} x^3-\frac{8735}{21} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{79411}{63} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*(5956 + 14691*x^2)*Sqrt[2 + x^2 - x^4])/63 - (1825*x*(2 + x^2 - x^4)^(3/2))/2
1 - (125*x^3*(2 + x^2 - x^4)^(3/2))/9 + (79411*EllipticE[ArcSin[x/Sqrt[2]], -2])
/63 - (8735*EllipticF[ArcSin[x/Sqrt[2]], -2])/21

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Rubi [A]  time = 0.22413, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{1825}{21} \left (-x^4+x^2+2\right )^{3/2} x+\frac{1}{63} \left (14691 x^2+5956\right ) \sqrt{-x^4+x^2+2} x-\frac{125}{9} \left (-x^4+x^2+2\right )^{3/2} x^3-\frac{8735}{21} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{79411}{63} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(5956 + 14691*x^2)*Sqrt[2 + x^2 - x^4])/63 - (1825*x*(2 + x^2 - x^4)^(3/2))/2
1 - (125*x^3*(2 + x^2 - x^4)^(3/2))/9 + (79411*EllipticE[ArcSin[x/Sqrt[2]], -2])
/63 - (8735*EllipticF[ArcSin[x/Sqrt[2]], -2])/21

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Rubi in Sympy [A]  time = 42.0977, size = 94, normalized size = 0.99 \[ - \frac{125 x^{3} \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{9} + \frac{x \left (\frac{24485 x^{2}}{7} + \frac{29780}{21}\right ) \sqrt{- x^{4} + x^{2} + 2}}{15} - \frac{1825 x \left (- x^{4} + x^{2} + 2\right )^{\frac{3}{2}}}{21} + \frac{79411 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{63} - \frac{8735 F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{21} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5*x**2+7)**3*(-x**4+x**2+2)**(1/2),x)

[Out]

-125*x**3*(-x**4 + x**2 + 2)**(3/2)/9 + x*(24485*x**2/7 + 29780/21)*sqrt(-x**4 +
x**2 + 2)/15 - 1825*x*(-x**4 + x**2 + 2)**(3/2)/21 + 79411*elliptic_e(asin(sqrt
(2)*x/2), -2)/63 - 8735*elliptic_f(asin(sqrt(2)*x/2), -2)/21

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Mathematica [C]  time = 0.104221, size = 107, normalized size = 1.13 \[ \frac{-875 x^{11}-3725 x^9-1116 x^7+21660 x^5+9938 x^3-106014 i \sqrt{-2 x^4+2 x^2+4} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+79411 i \sqrt{-2 x^4+2 x^2+4} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-9988 x}{63 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-9988*x + 9938*x^3 + 21660*x^5 - 1116*x^7 - 3725*x^9 - 875*x^11 + (79411*I)*Sqr
t[4 + 2*x^2 - 2*x^4]*EllipticE[I*ArcSinh[x], -1/2] - (106014*I)*Sqrt[4 + 2*x^2 -
2*x^4]*EllipticF[I*ArcSinh[x], -1/2])/(63*Sqrt[2 + x^2 - x^4])

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Maple [B]  time = 0.011, size = 176, normalized size = 1.9 \[ -{\frac{4994\,x}{63}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{26603\,\sqrt{2}}{63}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{79411\,\sqrt{2}}{126}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1} \left ({\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ) \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{7466\,{x}^{3}}{63}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{4600\,{x}^{5}}{63}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{125\,{x}^{7}}{9}\sqrt{-{x}^{4}+{x}^{2}+2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5*x^2+7)^3*(-x^4+x^2+2)^(1/2),x)

[Out]

-4994/63*x*(-x^4+x^2+2)^(1/2)+26603/63*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))-79411/126*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))-Elli
pticE(1/2*2^(1/2)*x,I*2^(1/2)))+7466/63*x^3*(-x^4+x^2+2)^(1/2)+4600/63*x^5*(-x^4
+x^2+2)^(1/2)+125/9*x^7*(-x^4+x^2+2)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343\right )} \sqrt{-x^{4} + x^{2} + 2}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((125*x^6 + 525*x^4 + 735*x^2 + 343)*sqrt(-x^4 + x^2 + 2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x**2+7)**3*(-x**4+x**2+2)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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