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

Optimal. Leaf size=93 \[ -\frac{17 \sqrt{-x^4+x^2+2} x}{175 \left (5 x^2+7\right )}-\frac{1}{75} \sqrt{-x^4+x^2+2} x+\frac{458}{875} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{97}{525} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{1241 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{6125} \]

[Out]

-(x*Sqrt[2 + x^2 - x^4])/75 - (17*x*Sqrt[2 + x^2 - x^4])/(175*(7 + 5*x^2)) - (97
*EllipticE[ArcSin[x/Sqrt[2]], -2])/525 + (458*EllipticF[ArcSin[x/Sqrt[2]], -2])/
875 - (1241*EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -2])/6125

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Rubi [A]  time = 0.722154, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{17 \sqrt{-x^4+x^2+2} x}{175 \left (5 x^2+7\right )}-\frac{1}{75} \sqrt{-x^4+x^2+2} x+\frac{458}{875} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{97}{525} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{1241 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{6125} \]

Antiderivative was successfully verified.

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

[Out]

-(x*Sqrt[2 + x^2 - x^4])/75 - (17*x*Sqrt[2 + x^2 - x^4])/(175*(7 + 5*x^2)) - (97
*EllipticE[ArcSin[x/Sqrt[2]], -2])/525 + (458*EllipticF[ArcSin[x/Sqrt[2]], -2])/
875 - (1241*EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -2])/6125

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Rubi in Sympy [A]  time = 114.061, size = 241, normalized size = 2.59 \[ - \frac{x \sqrt{- x^{4} + x^{2} + 2}}{75} - \frac{578 x \sqrt{- x^{4} + x^{2} + 2}}{25 \left (1190 x^{2} + 1666\right )} - \frac{97 \sqrt{2} \sqrt{- x^{4} + x^{2} + 2} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{1050 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} - \frac{209 \sqrt{2} \sqrt{- x^{4} + x^{2} + 2} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{8750 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} - \frac{1207 \sqrt{2} \sqrt{- x^{4} + x^{2} + 2} \Pi \left (- \frac{10}{7}; \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{61250 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} + \frac{51 \sqrt{- x^{4} + x^{2} + 2} \Pi \left (\frac{2}{7}; \operatorname{atan}{\left (x \right )}\middle | \frac{3}{2}\right )}{250 \sqrt{\frac{- \frac{x^{2}}{2} + 1}{x^{2} + 1}} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*sqrt(-x**4 + x**2 + 2)/75 - 578*x*sqrt(-x**4 + x**2 + 2)/(25*(1190*x**2 + 166
6)) - 97*sqrt(2)*sqrt(-x**4 + x**2 + 2)*elliptic_e(asin(sqrt(2)*x/2), -2)/(1050*
sqrt(-x**2/2 + 1)*sqrt(x**2 + 1)) - 209*sqrt(2)*sqrt(-x**4 + x**2 + 2)*elliptic_
f(asin(sqrt(2)*x/2), -2)/(8750*sqrt(-x**2/2 + 1)*sqrt(x**2 + 1)) - 1207*sqrt(2)*
sqrt(-x**4 + x**2 + 2)*elliptic_pi(-10/7, asin(sqrt(2)*x/2), -2)/(61250*sqrt(-x*
*2/2 + 1)*sqrt(x**2 + 1)) + 51*sqrt(-x**4 + x**2 + 2)*elliptic_pi(2/7, atan(x),
3/2)/(250*sqrt((-x**2/2 + 1)/(x**2 + 1))*(x**2 + 1))

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Mathematica [C]  time = 0.292483, size = 201, normalized size = 2.16 \[ \frac{2450 x^7+4550 x^5-11900 x^3+567 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-6790 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+18615 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )+26061 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )-14000 x}{36750 \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(-14000*x - 11900*x^3 + 4550*x^5 + 2450*x^7 - (6790*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[
2 + x^2 - x^4]*EllipticE[I*ArcSinh[x], -1/2] + (567*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[
2 + x^2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] + (26061*I)*Sqrt[2]*Sqrt[2 + x^2 -
x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] + (18615*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 -
x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(36750*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]
)

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Maple [B]  time = 0.028, size = 180, normalized size = 1.9 \[ -{\frac{17\,x}{875\,{x}^{2}+1225}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{x}{75}\sqrt{-{x}^{4}+{x}^{2}+2}}+{\frac{229\,\sqrt{2}}{875}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{97\,\sqrt{2}}{1050}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{1241\,\sqrt{2}}{6125}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-17/175*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)-1/75*x*(-x^4+x^2+2)^(1/2)+229/875*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))-97/1050*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*Elliptic
E(1/2*2^(1/2)*x,I*2^(1/2))-1241/6125*2^(1/2)*(1-1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(-x
^4+x^2+2)^(1/2)*EllipticPi(1/2*2^(1/2)*x,-10/7,I*2^(1/2))

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

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. \[{\rm integral}\left (\frac{{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}}{25 \, x^{4} + 70 \, x^{2} + 49}, x\right ) \]

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((-x^4 + x^2 + 2)^(3/2)/(25*x^4 + 70*x^2 + 49), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**4+x**2+2)**(3/2)/(5*x**2+7)**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. \[ \int \frac{{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}}{{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]

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)