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

Optimal. Leaf size=381 \[ \frac{\sqrt{2 x^4+2 x^2+1} x}{\sqrt{2} \left (\sqrt{2} x^2+1\right )}+\frac{1}{4} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )+\frac{2^{3/4} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{\left (3 \sqrt{2}-2\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{5 \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{12\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}} \]

[Out]

(x*Sqrt[1 + 2*x^2 + 2*x^4])/(Sqrt[2]*(1 + Sqrt[2]*x^2)) + (Sqrt[5/3]*ArcTan[(Sqr
t[5/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/4 - ((1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x
^4)/(1 + Sqrt[2]*x^2)^2]*EllipticE[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(2^(3/
4)*Sqrt[1 + 2*x^2 + 2*x^4]) + (2^(3/4)*(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])/((-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 - Sqrt[2])/4])/(12*2^(3/4)*(2 - 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4
])

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Rubi [A]  time = 0.399423, antiderivative size = 483, normalized size of antiderivative = 1.27, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{2 x^4+2 x^2+1} x}{\sqrt{2} \left (\sqrt{2} x^2+1\right )}+\frac{1}{4} \sqrt{\frac{5}{3}} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )-\frac{\left (1-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{4 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{5 \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{2\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{5 \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{12\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*Sqrt[1 + 2*x^2 + 2*x^4])/(Sqrt[2]*(1 + Sqrt[2]*x^2)) + (Sqrt[5/3]*ArcTan[(Sqr
t[5/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/4 - ((1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x
^4)/(1 + Sqrt[2]*x^2)^2]*EllipticE[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(2^(3/
4)*Sqrt[1 + 2*x^2 + 2*x^4]) - (5*(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])/(2*2^(3/4)*(2
- 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4]) - ((1 - Sqrt[2])*(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])/(4*2^(1/4)*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 - Sqrt[2])/4])/(12*2^(3/4)*(2 - 3*Sqrt[2])*Sqrt[1 + 2*x^
2 + 2*x^4])

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Rubi in Sympy [A]  time = 45.0291, size = 432, normalized size = 1.13 \[ \frac{\sqrt{2} x \sqrt{2 x^{4} + 2 x^{2} + 1}}{2 \left (\sqrt{2} x^{2} + 1\right )} - \frac{\sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{2 \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{\sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (- 2 \sqrt{2} + 4\right ) \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{16 \sqrt{2 x^{4} + 2 x^{2} + 1}} - \frac{5 \sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{4 \left (- 3 \sqrt{2} + 2\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{5 \cdot 2^{\frac{3}{4}} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (2 + 3 \sqrt{2}\right ) \left (\sqrt{2} x^{2} + 1\right ) \Pi \left (- \frac{11 \sqrt{2}}{24} + \frac{1}{2}; 2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{48 \left (- 3 \sqrt{2} + 2\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{\sqrt{15} \operatorname{atan}{\left (\frac{\sqrt{15} x}{3 \sqrt{2 x^{4} + 2 x^{2} + 1}} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*x*sqrt(2*x**4 + 2*x**2 + 1)/(2*(sqrt(2)*x**2 + 1)) - 2**(1/4)*sqrt((2*x*
*4 + 2*x**2 + 1)/(sqrt(2)*x**2 + 1)**2)*(sqrt(2)*x**2 + 1)*elliptic_e(2*atan(2**
(1/4)*x), -sqrt(2)/4 + 1/2)/(2*sqrt(2*x**4 + 2*x**2 + 1)) + 2**(1/4)*sqrt((2*x**
4 + 2*x**2 + 1)/(sqrt(2)*x**2 + 1)**2)*(-2*sqrt(2) + 4)*(sqrt(2)*x**2 + 1)*ellip
tic_f(2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(16*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)*ell
iptic_f(2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(4*(-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)/(48*(-3*sqrt(2) + 2)*sqrt(2*x**4 + 2*x**2 + 1)) + sqrt(15
)*atan(sqrt(15)*x/(3*sqrt(2*x**4 + 2*x**2 + 1)))/12

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Mathematica [C]  time = 0.101229, size = 127, normalized size = 0.33 \[ -\frac{\sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \left (-(3+6 i) F\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+(3+3 i) E\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+5 i \Pi \left (\frac{1}{3}+\frac{i}{3};\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )\right )}{6 \sqrt{1-i} \sqrt{2 x^4+2 x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*((3 + 3*I)*EllipticE[I*ArcSinh[Sqr
t[1 - I]*x], I] - (3 + 6*I)*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + (5*I)*Ellip
ticPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I]))/(6*Sqrt[1 - I]*Sqrt[1 + 2*x^2 +
2*x^4])

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Maple [C]  time = 0.007, size = 341, normalized size = 0.9 \[ -{\frac{{\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{{\frac{i}{2}}{\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{2\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{{\frac{i}{2}}{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{5}{6\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\it EllipticPi} \left ( x\sqrt{-1+i},{\frac{1}{3}}+{\frac{i}{3}},{\frac{\sqrt{-1-i}}{\sqrt{-1+i}}} \right ){\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(-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)*E
llipticF(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))+1/2*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))+1/2/(-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))-1/2
*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)*E
llipticE(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))+5/6/(-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. \[ \int \frac{\sqrt{2 \, x^{4} + 2 \, x^{2} + 1}}{2 \, x^{2} + 3}\,{d x} \]

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, algorithm="maxima")

[Out]

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

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

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, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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, algorithm="giac")

[Out]

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