3.349 \(\int \frac{1}{\left (7+5 x^2\right ) \left (2+x^2-x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=72 \[ \frac{x \left (35-16 x^2\right )}{306 \sqrt{-x^4+x^2+2}}+\frac{1}{102} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{8}{153} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{25}{238} \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

[Out]

(x*(35 - 16*x^2))/(306*Sqrt[2 + x^2 - x^4]) + (8*EllipticE[ArcSin[x/Sqrt[2]], -2
])/153 + EllipticF[ArcSin[x/Sqrt[2]], -2]/102 - (25*EllipticPi[-10/7, ArcSin[x/S
qrt[2]], -2])/238

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Rubi [A]  time = 0.296828, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{x \left (35-16 x^2\right )}{306 \sqrt{-x^4+x^2+2}}+\frac{1}{102} F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )+\frac{8}{153} E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )-\frac{25}{238} \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x*(35 - 16*x^2))/(306*Sqrt[2 + x^2 - x^4]) + (8*EllipticE[ArcSin[x/Sqrt[2]], -2
])/153 + EllipticF[ArcSin[x/Sqrt[2]], -2]/102 - (25*EllipticPi[-10/7, ArcSin[x/S
qrt[2]], -2])/238

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Rubi in Sympy [A]  time = 81.7535, size = 73, normalized size = 1.01 \[ \frac{x \left (- 32 x^{2} + 70\right )}{612 \sqrt{- x^{4} + x^{2} + 2}} + \frac{8 E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{153} + \frac{F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{102} - \frac{25 \Pi \left (- \frac{10}{7}; \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{238} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(-32*x**2 + 70)/(612*sqrt(-x**4 + x**2 + 2)) + 8*elliptic_e(asin(sqrt(2)*x/2),
-2)/153 + elliptic_f(asin(sqrt(2)*x/2), -2)/102 - 25*elliptic_pi(-10/7, asin(sq
rt(2)*x/2), -2)/238

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Mathematica [C]  time = 0.212783, size = 101, normalized size = 1.4 \[ \frac{\frac{490 x}{\sqrt{-x^4+x^2+2}}-\frac{224 x^3}{\sqrt{-x^4+x^2+2}}-357 i \sqrt{2} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+224 i \sqrt{2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+225 i \sqrt{2} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )}{4284} \]

Antiderivative was successfully verified.

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

[Out]

((490*x)/Sqrt[2 + x^2 - x^4] - (224*x^3)/Sqrt[2 + x^2 - x^4] + (224*I)*Sqrt[2]*E
llipticE[I*ArcSinh[x], -1/2] - (357*I)*Sqrt[2]*EllipticF[I*ArcSinh[x], -1/2] + (
225*I)*Sqrt[2]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/4284

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Maple [B]  time = 0.023, size = 164, normalized size = 2.3 \[ 2\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ( -{\frac{4\,{x}^{3}}{153}}+{\frac{35\,x}{612}} \right ) }+{\frac{\sqrt{2}}{204}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{4\,\sqrt{2}}{153}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{25\,\sqrt{2}}{238}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(-4/153*x^3+35/612*x)/(-x^4+x^2+2)^(1/2)+1/204*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))+4/153*2^(1/2)*(-2*
x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*EllipticE(1/2*2^(1/2)*x,I*2^(1/2))
-25/238*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. \[ \int \frac{1}{{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)),x, algorithm="maxima")

[Out]

integrate(1/((-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. \[{\rm integral}\left (-\frac{1}{{\left (5 \, x^{6} + 2 \, x^{4} - 17 \, x^{2} - 14\right )} \sqrt{-x^{4} + x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)),x, algorithm="fricas")

[Out]

integral(-1/((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. \[ \int \frac{1}{\left (- \left (x^{2} - 2\right ) \left (x^{2} + 1\right )\right )^{\frac{3}{2}} \left (5 x^{2} + 7\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((-(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. \[ \int \frac{1}{{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)),x, algorithm="giac")

[Out]

integrate(1/((-x^4 + x^2 + 2)^(3/2)*(5*x^2 + 7)), x)