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

Optimal. Leaf size=102 \[ -\frac{31 \sqrt{-x^4+x^2+2} x}{13328 \left (5 x^2+7\right )}+\frac{\sqrt{-x^4+x^2+2} x}{28 \left (5 x^2+7\right )^2}-\frac{269 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{166600}-\frac{31 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{66640}+\frac{16601 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{2332400} \]

[Out]

(x*Sqrt[2 + x^2 - x^4])/(28*(7 + 5*x^2)^2) - (31*x*Sqrt[2 + x^2 - x^4])/(13328*(
7 + 5*x^2)) - (31*EllipticE[ArcSin[x/Sqrt[2]], -2])/66640 - (269*EllipticF[ArcSi
n[x/Sqrt[2]], -2])/166600 + (16601*EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -2])/233
2400

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Rubi [A]  time = 0.891212, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 11, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.458 \[ -\frac{31 \sqrt{-x^4+x^2+2} x}{13328 \left (5 x^2+7\right )}+\frac{\sqrt{-x^4+x^2+2} x}{28 \left (5 x^2+7\right )^2}-\frac{269 F\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{166600}-\frac{31 E\left (\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{66640}+\frac{16601 \Pi \left (-\frac{10}{7};\left .\sin ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |-2\right )}{2332400} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[2 + x^2 - x^4])/(28*(7 + 5*x^2)^2) - (31*x*Sqrt[2 + x^2 - x^4])/(13328*(
7 + 5*x^2)) - (31*EllipticE[ArcSin[x/Sqrt[2]], -2])/66640 - (269*EllipticF[ArcSi
n[x/Sqrt[2]], -2])/166600 + (16601*EllipticPi[-10/7, ArcSin[x/Sqrt[2]], -2])/233
2400

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [C]  time = 0.467603, size = 244, normalized size = 2.39 \[ \frac{54250 x^7-144900 x^5-17850 x^3+7021 i \sqrt{2} \left (5 x^2+7\right )^2 \sqrt{-x^4+x^2+2} F\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-2170 i \sqrt{2} \left (5 x^2+7\right )^2 \sqrt{-x^4+x^2+2} E\left (i \sinh ^{-1}(x)|-\frac{1}{2}\right )-415025 i \sqrt{2} \sqrt{-x^4+x^2+2} x^4 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )-1162070 i \sqrt{2} \sqrt{-x^4+x^2+2} x^2 \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )-813449 i \sqrt{2} \sqrt{-x^4+x^2+2} \Pi \left (\frac{5}{7};i \sinh ^{-1}(x)|-\frac{1}{2}\right )+181300 x}{4664800 \left (5 x^2+7\right )^2 \sqrt{-x^4+x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(181300*x - 17850*x^3 - 144900*x^5 + 54250*x^7 - (2170*I)*Sqrt[2]*(7 + 5*x^2)^2*
Sqrt[2 + x^2 - x^4]*EllipticE[I*ArcSinh[x], -1/2] + (7021*I)*Sqrt[2]*(7 + 5*x^2)
^2*Sqrt[2 + x^2 - x^4]*EllipticF[I*ArcSinh[x], -1/2] - (813449*I)*Sqrt[2]*Sqrt[2
+ x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] - (1162070*I)*Sqrt[2]*x^2*Sqrt
[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2] - (415025*I)*Sqrt[2]*x^4*Sqr
t[2 + x^2 - x^4]*EllipticPi[5/7, I*ArcSinh[x], -1/2])/(4664800*(7 + 5*x^2)^2*Sqr
t[2 + x^2 - x^4])

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Maple [A]  time = 0.029, size = 189, normalized size = 1.9 \[{\frac{x}{28\, \left ( 5\,{x}^{2}+7 \right ) ^{2}}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{31\,x}{66640\,{x}^{2}+93296}\sqrt{-{x}^{4}+{x}^{2}+2}}-{\frac{269\,\sqrt{2}}{333200}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticF} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}-{\frac{31\,\sqrt{2}}{133280}\sqrt{-2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\it EllipticE} \left ({\frac{\sqrt{2}x}{2}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}}+{\frac{16601\,\sqrt{2}}{2332400}\sqrt{1-{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{\sqrt{2}x}{2}},-{\frac{10}{7}},i\sqrt{2} \right ){\frac{1}{\sqrt{-{x}^{4}+{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/28*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)^2-31/13328*x*(-x^4+x^2+2)^(1/2)/(5*x^2+7)-26
9/333200*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))-31/133280*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))+16601/2332400*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. \[ \int \frac{\sqrt{-x^{4} + x^{2} + 2}}{{\left (5 \, x^{2} + 7\right )}^{3}}\,{d x} \]

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. \[{\rm integral}\left (\frac{\sqrt{-x^{4} + x^{2} + 2}}{125 \, x^{6} + 525 \, x^{4} + 735 \, x^{2} + 343}, x\right ) \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

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)