3.312 \(\int \frac{\sqrt{1+2 x^2+2 x^4}}{x^4 \left (3+2 x^2\right )} \, dx\)

Optimal. Leaf size=373 \[ \frac{1}{9} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )-\frac{5 \sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}+\frac{5 \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{27\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt{2 x^4+2 x^2+1}}{9 x^3} \]

[Out]

-Sqrt[1 + 2*x^2 + 2*x^4]/(9*x^3) + (Sqrt[5/3]*ArcTan[(Sqrt[5/3]*x)/Sqrt[1 + 2*x^
2 + 2*x^4]])/9 - ((1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2
]*EllipticF[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(9*2^(1/4)*Sqrt[1 + 2*x^2 + 2
*x^4]) - (5*2^(1/4)*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)
^2]*EllipticF[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(9*(2 - 3*Sqrt[2])*Sqrt[1 +
2*x^2 + 2*x^4]) + (5*(3 + Sqrt[2])*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(
1 + Sqrt[2]*x^2)^2]*EllipticPi[(12 - 11*Sqrt[2])/24, 2*ArcTan[2^(1/4)*x], (2 - S
qrt[2])/4])/(27*2^(3/4)*(2 - 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4])

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Rubi [A]  time = 0.392102, antiderivative size = 373, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{1}{9} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )-\frac{5 \sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{9 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}+\frac{5 \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{27\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\sqrt{2 x^4+2 x^2+1}}{9 x^3} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[1 + 2*x^2 + 2*x^4]/(9*x^3) + (Sqrt[5/3]*ArcTan[(Sqrt[5/3]*x)/Sqrt[1 + 2*x^
2 + 2*x^4]])/9 - ((1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2
]*EllipticF[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(9*2^(1/4)*Sqrt[1 + 2*x^2 + 2
*x^4]) - (5*2^(1/4)*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)
^2]*EllipticF[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(9*(2 - 3*Sqrt[2])*Sqrt[1 +
2*x^2 + 2*x^4]) + (5*(3 + Sqrt[2])*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(
1 + Sqrt[2]*x^2)^2]*EllipticPi[(12 - 11*Sqrt[2])/24, 2*ArcTan[2^(1/4)*x], (2 - S
qrt[2])/4])/(27*2^(3/4)*(2 - 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4])

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Rubi in Sympy [A]  time = 37.1327, size = 330, normalized size = 0.88 \[ - \frac{2^{\frac{3}{4}} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{18 \sqrt{2 x^{4} + 2 x^{2} + 1}} - \frac{5 \sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{9 \left (- 3 \sqrt{2} + 2\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{5 \cdot 2^{\frac{3}{4}} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (2 + 3 \sqrt{2}\right ) \left (\sqrt{2} x^{2} + 1\right ) \Pi \left (- \frac{11 \sqrt{2}}{24} + \frac{1}{2}; 2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{108 \left (- 3 \sqrt{2} + 2\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{\sqrt{15} \operatorname{atan}{\left (\frac{\sqrt{15} x}{3 \sqrt{2 x^{4} + 2 x^{2} + 1}} \right )}}{27} - \frac{\sqrt{2 x^{4} + 2 x^{2} + 1}}{9 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2**(3/4)*sqrt((2*x**4 + 2*x**2 + 1)/(sqrt(2)*x**2 + 1)**2)*(sqrt(2)*x**2 + 1)*e
lliptic_f(2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(18*sqrt(2*x**4 + 2*x**2 + 1)) -
5*2**(1/4)*sqrt((2*x**4 + 2*x**2 + 1)/(sqrt(2)*x**2 + 1)**2)*(sqrt(2)*x**2 + 1)
*elliptic_f(2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(9*(-3*sqrt(2) + 2)*sqrt(2*x**
4 + 2*x**2 + 1)) + 5*2**(3/4)*sqrt((2*x**4 + 2*x**2 + 1)/(sqrt(2)*x**2 + 1)**2)*
(2 + 3*sqrt(2))*(sqrt(2)*x**2 + 1)*elliptic_pi(-11*sqrt(2)/24 + 1/2, 2*atan(2**(
1/4)*x), -sqrt(2)/4 + 1/2)/(108*(-3*sqrt(2) + 2)*sqrt(2*x**4 + 2*x**2 + 1)) + sq
rt(15)*atan(sqrt(15)*x/(3*sqrt(2*x**4 + 2*x**2 + 1)))/27 - sqrt(2*x**4 + 2*x**2
+ 1)/(9*x**3)

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Mathematica [C]  time = 0.165498, size = 154, normalized size = 0.41 \[ -\frac{6 x^4+6 x^2+3 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 F\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )-5 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^3 \Pi \left (\frac{1}{3}+\frac{i}{3};\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+3}{27 x^3 \sqrt{2 x^4+2 x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

-(3 + 6*x^2 + 6*x^4 + 3*(1 - I)^(3/2)*x^3*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)
*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] - 5*(1 - I)^(3/2)*x^3*Sqrt[1 + (1 -
I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I
])/(27*x^3*Sqrt[1 + 2*x^2 + 2*x^4])

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Maple [C]  time = 0.024, size = 448, normalized size = 1.2 \[ -{\frac{1}{9\,{x}^{3}}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}+{\frac{ \left ({\frac{2}{9}}-{\frac{2\,i}{9}} \right ) \left ({\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) -{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) \right ) }{\sqrt{-1+i}}\sqrt{1+ \left ( 1-i \right ){x}^{2}}\sqrt{1+ \left ( 1+i \right ){x}^{2}}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{4\,{\it EllipticF} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{9\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{{\frac{2\,i}{9}}{\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{2\,{\it EllipticE} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{9\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{{\frac{2\,i}{9}}{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{10}{27\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\it EllipticPi} \left ( x\sqrt{-1+i},{\frac{1}{3}}+{\frac{i}{3}},{\frac{\sqrt{-1-i}}{\sqrt{-1+i}}} \right ){\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*(2*x^4+2*x^2+1)^(1/2)/x^3+(2/9-2/9*I)/(-1+I)^(1/2)*(1+(1-I)*x^2)^(1/2)*(1+(
1+I)*x^2)^(1/2)/(2*x^4+2*x^2+1)^(1/2)*(EllipticF(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*
I*2^(1/2))-EllipticE(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2)))-4/9/(-1+I)^(1/2)
*(-I*x^2+x^2+1)^(1/2)*(I*x^2+x^2+1)^(1/2)/(2*x^4+2*x^2+1)^(1/2)*EllipticF(x*(-1+
I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))+2/9*I/(-1+I)^(1/2)*(-I*x^2+x^2+1)^(1/2)*(I*x
^2+x^2+1)^(1/2)/(2*x^4+2*x^2+1)^(1/2)*EllipticF(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I
*2^(1/2))+2/9/(-1+I)^(1/2)*(-I*x^2+x^2+1)^(1/2)*(I*x^2+x^2+1)^(1/2)/(2*x^4+2*x^2
+1)^(1/2)*EllipticE(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))-2/9*I/(-1+I)^(1/2)
*(-I*x^2+x^2+1)^(1/2)*(I*x^2+x^2+1)^(1/2)/(2*x^4+2*x^2+1)^(1/2)*EllipticE(x*(-1+
I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))+10/27/(-1+I)^(1/2)*(-I*x^2+x^2+1)^(1/2)*(I*x
^2+x^2+1)^(1/2)/(2*x^4+2*x^2+1)^(1/2)*EllipticPi(x*(-1+I)^(1/2),1/3+1/3*I,(-1-I)
^(1/2)/(-1+I)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 \, x^{4} + 2 \, x^{2} + 1}}{{\left (2 \, x^{2} + 3\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(2*x^4 + 2*x^2 + 1)/((2*x^2 + 3)*x^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{2 \, x^{4} + 2 \, x^{2} + 1}}{2 \, x^{6} + 3 \, x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(2*x^4 + 2*x^2 + 1)/(2*x^6 + 3*x^4), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 x^{4} + 2 x^{2} + 1}}{x^{4} \left (2 x^{2} + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(2*x**4 + 2*x**2 + 1)/(x**4*(2*x**2 + 3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{2 \, x^{4} + 2 \, x^{2} + 1}}{{\left (2 \, x^{2} + 3\right )} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(2*x^4 + 2*x^2 + 1)/((2*x^2 + 3)*x^4), x)