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

Optimal. Leaf size=100 \[ \frac{625 \sqrt{-x^4+x^2+2} x}{16184 \left (5 x^2+7\right )}+\frac{\left (580-287 x^2\right ) x}{10404 \sqrt{-x^4+x^2+2}}+\frac{89 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{24276}+\frac{5143 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{145656}-\frac{10825 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{113288} \]

[Out]

(x*(580 - 287*x^2))/(10404*Sqrt[2 + x^2 - x^4]) + (625*x*Sqrt[2 + x^2 - x^4])/(1
6184*(7 + 5*x^2)) + (5143*EllipticE[ArcSin[x/Sqrt[2]], -2])/145656 + (89*Ellipti
cF[ArcSin[x/Sqrt[2]], -2])/24276 - (10825*EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -
2])/113288

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Rubi [A]  time = 0.662267, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375 \[ \frac{625 \sqrt{-x^4+x^2+2} x}{16184 \left (5 x^2+7\right )}+\frac{\left (580-287 x^2\right ) x}{10404 \sqrt{-x^4+x^2+2}}+\frac{89 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{24276}+\frac{5143 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{145656}-\frac{10825 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{113288} \]

Antiderivative was successfully verified.

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

[Out]

(x*(580 - 287*x^2))/(10404*Sqrt[2 + x^2 - x^4]) + (625*x*Sqrt[2 + x^2 - x^4])/(1
6184*(7 + 5*x^2)) + (5143*EllipticE[ArcSin[x/Sqrt[2]], -2])/145656 + (89*Ellipti
cF[ArcSin[x/Sqrt[2]], -2])/24276 - (10825*EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -
2])/113288

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Rubi in Sympy [A]  time = 144.434, size = 299, normalized size = 2.99 \[ \frac{625 x \sqrt{- x^{4} + x^{2} + 2}}{68 \left (1190 x^{2} + 1666\right )} + \frac{1595 x \sqrt{- x^{4} + x^{2} + 2}}{58956 \left (x^{2} + 1\right )} - \frac{47 x \sqrt{- x^{4} + x^{2} + 2}}{176868 \left (- \frac{x^{2}}{2} + 1\right )} + \frac{16 x \sqrt{- x^{4} + x^{2} + 2}}{14739 \left (- \frac{x^{2}}{2} + 1\right ) \left (x^{2} + 1\right )} + \frac{5143 \sqrt{2} \sqrt{- x^{4} + x^{2} + 2} E\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{291312 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} - \frac{9797 \sqrt{2} \sqrt{- x^{4} + x^{2} + 2} F\left (\operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{97104 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} - \frac{4175 \sqrt{2} \sqrt{- x^{4} + x^{2} + 2} \Pi \left (- \frac{10}{7}; \operatorname{asin}{\left (\frac{\sqrt{2} x}{2} \right )}\middle | -2\right )}{226576 \sqrt{- \frac{x^{2}}{2} + 1} \sqrt{x^{2} + 1}} + \frac{2375 \sqrt{- x^{4} + x^{2} + 2} \Pi \left (\frac{2}{7}; \operatorname{atan}{\left (x \right )}\middle | \frac{3}{2}\right )}{32368 \sqrt{\frac{- \frac{x^{2}}{2} + 1}{x^{2} + 1}} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625*x*sqrt(-x**4 + x**2 + 2)/(68*(1190*x**2 + 1666)) + 1595*x*sqrt(-x**4 + x**2
+ 2)/(58956*(x**2 + 1)) - 47*x*sqrt(-x**4 + x**2 + 2)/(176868*(-x**2/2 + 1)) + 1
6*x*sqrt(-x**4 + x**2 + 2)/(14739*(-x**2/2 + 1)*(x**2 + 1)) + 5143*sqrt(2)*sqrt(
-x**4 + x**2 + 2)*elliptic_e(asin(sqrt(2)*x/2), -2)/(291312*sqrt(-x**2/2 + 1)*sq
rt(x**2 + 1)) - 9797*sqrt(2)*sqrt(-x**4 + x**2 + 2)*elliptic_f(asin(sqrt(2)*x/2)
, -2)/(97104*sqrt(-x**2/2 + 1)*sqrt(x**2 + 1)) - 4175*sqrt(2)*sqrt(-x**4 + x**2
+ 2)*elliptic_pi(-10/7, asin(sqrt(2)*x/2), -2)/(226576*sqrt(-x**2/2 + 1)*sqrt(x*
*2 + 1)) + 2375*sqrt(-x**4 + x**2 + 2)*elliptic_pi(2/7, atan(x), 3/2)/(32368*sqr
t((-x**2/2 + 1)/(x**2 + 1))*(x**2 + 1))

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Mathematica [C]  time = 0.292675, size = 196, normalized size = 1.96 \[ \frac{-360010 x^5+253386 x^3-111741 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+72002 i \sqrt{2} \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )+487125 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )+681975 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )+953260 x}{2039184 \left (5 x^2+7\right ) \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(953260*x + 253386*x^3 - 360010*x^5 + (72002*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 + x^2
- x^4]*EllipticE[I*ArcSinh[x], -1/2] - (111741*I)*Sqrt[2]*(7 + 5*x^2)*Sqrt[2 +
x^2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] + (681975*I)*Sqrt[2]*Sqrt[2 + x^2 - x^4
]*EllipticPi[5/7, I*ArcSinh[x], -1/2] + (487125*I)*Sqrt[2]*x^2*Sqrt[2 + x^2 - x^
4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(2039184*(7 + 5*x^2)*Sqrt[2 + x^2 - x^4]
)

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Maple [B]  time = 0.03, size = 188, normalized size = 1.9 \[{\frac{625\,x}{80920\,{x}^{2}+113288}\sqrt{-{x}^{4}+{x}^{2}+2}}+2\,{\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}} \left ( -{\frac{287\,{x}^{3}}{20808}}+{\frac{145\,x}{5202}} \right ) }+{\frac{89\,\sqrt{2}}{48552}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{5143\,\sqrt{2}}{291312}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{10825\,\sqrt{2}}{113288}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

625/16184*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)+2*(-287/20808*x^3+145/5202*x)/(-x^4+x^2
+2)^(1/2)+89/48552*2^(1/2)*(-2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(-x^4+x^2+2)^(1/2)*Ell
ipticF(1/2*2^(1/2)*x,I*2^(1/2))+5143/291312*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))-10825/113288*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. \[ \int \frac{1}{{\left (-x^{4} + x^{2} + 2\right )}^{\frac{3}{2}}{\left (5 \, x^{2} + 7\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((-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. \[{\rm integral}\left (-\frac{1}{{\left (25 \, x^{8} + 45 \, x^{6} - 71 \, x^{4} - 189 \, x^{2} - 98\right )} \sqrt{-x^{4} + x^{2} + 2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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