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

Optimal. Leaf size=209 \[ \frac{\sqrt{x^4+3 x^2+2} x}{14 \left (5 x^2+7\right )}-\frac{\left (x^2+2\right ) x}{70 \sqrt{x^4+3 x^2+2}}+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{140 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{35 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{\left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{980 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

[Out]

-(x*(2 + x^2))/(70*Sqrt[2 + 3*x^2 + x^4]) + (x*Sqrt[2 + 3*x^2 + x^4])/(14*(7 + 5
*x^2)) + ((1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(35*Sqr
t[2]*Sqrt[2 + 3*x^2 + x^4]) + (3*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[A
rcTan[x], 1/2])/(140*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) - ((2 + x^2)*EllipticPi[2/7,
ArcTan[x], 1/2])/(980*Sqrt[2]*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])

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Rubi [A]  time = 0.336011, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{\sqrt{x^4+3 x^2+2} x}{14 \left (5 x^2+7\right )}-\frac{\left (x^2+2\right ) x}{70 \sqrt{x^4+3 x^2+2}}+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{140 \sqrt{2} \sqrt{x^4+3 x^2+2}}+\frac{\left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{35 \sqrt{2} \sqrt{x^4+3 x^2+2}}-\frac{\left (x^2+2\right ) \Pi \left (\frac{2}{7};\tan ^{-1}(x)|\frac{1}{2}\right )}{980 \sqrt{2} \sqrt{\frac{x^2+2}{x^2+1}} \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

-(x*(2 + x^2))/(70*Sqrt[2 + 3*x^2 + x^4]) + (x*Sqrt[2 + 3*x^2 + x^4])/(14*(7 + 5
*x^2)) + ((1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticE[ArcTan[x], 1/2])/(35*Sqr
t[2]*Sqrt[2 + 3*x^2 + x^4]) + (3*(1 + x^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[A
rcTan[x], 1/2])/(140*Sqrt[2]*Sqrt[2 + 3*x^2 + x^4]) - ((2 + x^2)*EllipticPi[2/7,
ArcTan[x], 1/2])/(980*Sqrt[2]*Sqrt[(2 + x^2)/(1 + x^2)]*Sqrt[2 + 3*x^2 + x^4])

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Rubi in Sympy [A]  time = 52.7469, size = 173, normalized size = 0.83 \[ \frac{x \sqrt{x^{4} + 3 x^{2} + 2}}{70 x^{2} + 98} - \frac{x \sqrt{x^{4} + 3 x^{2} + 2}}{70 \left (x^{2} + 1\right )} + \frac{\sqrt{x^{4} + 3 x^{2} + 2} E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{70 \sqrt{\frac{\frac{x^{2}}{2} + 1}{x^{2} + 1}} \left (x^{2} + 1\right )} + \frac{3 \sqrt{x^{4} + 3 x^{2} + 2} F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{280 \sqrt{\frac{\frac{x^{2}}{2} + 1}{x^{2} + 1}} \left (x^{2} + 1\right )} - \frac{\sqrt{x^{4} + 3 x^{2} + 2} \Pi \left (\frac{2}{7}; \operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{1960 \sqrt{\frac{\frac{x^{2}}{2} + 1}{x^{2} + 1}} \left (x^{2} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*sqrt(x**4 + 3*x**2 + 2)/(70*x**2 + 98) - x*sqrt(x**4 + 3*x**2 + 2)/(70*(x**2 +
1)) + sqrt(x**4 + 3*x**2 + 2)*elliptic_e(atan(x), 1/2)/(70*sqrt((x**2/2 + 1)/(x
**2 + 1))*(x**2 + 1)) + 3*sqrt(x**4 + 3*x**2 + 2)*elliptic_f(atan(x), 1/2)/(280*
sqrt((x**2/2 + 1)/(x**2 + 1))*(x**2 + 1)) - sqrt(x**4 + 3*x**2 + 2)*elliptic_pi(
2/7, atan(x), 1/2)/(1960*sqrt((x**2/2 + 1)/(x**2 + 1))*(x**2 + 1))

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Mathematica [C]  time = 0.209846, size = 208, normalized size = 1. \[ \frac{175 x^5+525 x^3-84 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right ) F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+35 i \sqrt{x^2+1} \sqrt{x^2+2} \left (5 x^2+7\right ) E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-5 i \sqrt{x^2+1} \sqrt{x^2+2} x^2 \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-7 i \sqrt{x^2+1} \sqrt{x^2+2} \Pi \left (\frac{10}{7};\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+350 x}{2450 \left (5 x^2+7\right ) \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(350*x + 525*x^3 + 175*x^5 + (35*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)*Elli
pticE[I*ArcSinh[x/Sqrt[2]], 2] - (84*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*(7 + 5*x^2)*
EllipticF[I*ArcSinh[x/Sqrt[2]], 2] - (7*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*EllipticP
i[10/7, I*ArcSinh[x/Sqrt[2]], 2] - (5*I)*x^2*Sqrt[1 + x^2]*Sqrt[2 + x^2]*Ellipti
cPi[10/7, I*ArcSinh[x/Sqrt[2]], 2])/(2450*(7 + 5*x^2)*Sqrt[2 + 3*x^2 + x^4])

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Maple [C]  time = 0.027, size = 162, normalized size = 0.8 \[{\frac{x}{70\,{x}^{2}+98}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{3\,i}{175}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{i}{140}}\sqrt{2}{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}-{{\frac{i}{2450}}\sqrt{2}\sqrt{1+{\frac{{x}^{2}}{2}}}\sqrt{{x}^{2}+1}{\it EllipticPi} \left ({\frac{i}{2}}\sqrt{2}x,{\frac{10}{7}},\sqrt{2} \right ){\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/14*x*(x^4+3*x^2+2)^(1/2)/(5*x^2+7)-3/175*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/
2)/(x^4+3*x^2+2)^(1/2)*EllipticF(1/2*I*2^(1/2)*x,2^(1/2))+1/140*I*2^(1/2)*(2*x^2
+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticE(1/2*I*2^(1/2)*x,2^(1/2))-1
/2450*I*2^(1/2)*(1+1/2*x^2)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*EllipticPi(1
/2*I*2^(1/2)*x,10/7,2^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{x^{4} + 3 \, x^{2} + 2}}{25 \, x^{4} + 70 \, x^{2} + 49}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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