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

Optimal. Leaf size=398 \[ \frac{\sqrt{2 x^4+2 x^2+1} x}{2 \sqrt{2} \left (\sqrt{2} x^2+1\right )}+\frac{3}{8} \sqrt{\frac{3}{5}} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )-\frac{\left (\left (\sqrt{2}-6\right ) x^2-3 \sqrt{2}+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{2\ 2^{3/4} \left (3 \sqrt{2}-2\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{2\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{3 \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{8\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}} \]

[Out]

(x*Sqrt[1 + 2*x^2 + 2*x^4])/(2*Sqrt[2]*(1 + Sqrt[2]*x^2)) + (3*Sqrt[3/5]*ArcTan[
(Sqrt[5/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/8 - ((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
*2^(3/4)*Sqrt[1 + 2*x^2 + 2*x^4]) - ((1 - 3*Sqrt[2] + (-6 + 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]) + (3*(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])/(8*2^(3/4)*(2 - 3*Sqrt[
2])*Sqrt[1 + 2*x^2 + 2*x^4])

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Rubi [A]  time = 0.462023, antiderivative size = 488, normalized size of antiderivative = 1.23, number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ \frac{\sqrt{2 x^4+2 x^2+1} x}{2 \sqrt{2} \left (\sqrt{2} x^2+1\right )}+\frac{3}{8} \sqrt{\frac{3}{5}} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )-\frac{\left (3-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{8 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{9 \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{4\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{2\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{3 \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{8\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(x*Sqrt[1 + 2*x^2 + 2*x^4])/(2*Sqrt[2]*(1 + Sqrt[2]*x^2)) + (3*Sqrt[3/5]*ArcTan[
(Sqrt[5/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/8 - ((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
*2^(3/4)*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])/(4*2^(3/
4)*(2 - 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4]) - ((3 - Sqrt[2])*(1 + Sqrt[2]*x^2)*S
qrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2]*EllipticF[2*ArcTan[2^(1/4)*x], (2 -
Sqrt[2])/4])/(8*2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4]) + (3*(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])/(8*2^(3/4)*(2 - 3*Sqrt[2])*Sqrt[1 +
2*x^2 + 2*x^4])

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Rubi in Sympy [A]  time = 43.7509, size = 507, normalized size = 1.27 \[ \frac{\sqrt{2} x \sqrt{2 x^{4} + 2 x^{2} + 1}}{4 \left (\sqrt{2} x^{2} + 1\right )} - \frac{\sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{4 \sqrt{2 x^{4} + 2 x^{2} + 1}} - \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{16 \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{\sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{8 \sqrt{2 x^{4} + 2 x^{2} + 1}} - \frac{9 \sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (\sqrt{2} x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{8 \left (- 3 \sqrt{2} + 2\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{3 \cdot 2^{\frac{3}{4}} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (2 + 3 \sqrt{2}\right ) \left (\sqrt{2} x^{2} + 1\right ) \Pi \left (- \frac{11 \sqrt{2}}{24} + \frac{1}{2}; 2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{32 \left (- 3 \sqrt{2} + 2\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{3 \sqrt{15} \operatorname{atan}{\left (\frac{\sqrt{15} x}{3 \sqrt{2 x^{4} + 2 x^{2} + 1}} \right )}}{40} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*x*sqrt(2*x**4 + 2*x**2 + 1)/(4*(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)/(4*sqrt(2*x**4 + 2*x**2 + 1)) - 3*2**(3/4)*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)/(16*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)*(sqrt(2)*x**2 + 1)*elliptic_f(2*atan(2*
*(1/4)*x), -sqrt(2)/4 + 1/2)/(8*sqrt(2*x**4 + 2*x**2 + 1)) - 9*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)/(8*(-3*sqrt(2) + 2)*sqrt(2*x**4 + 2*x**2 + 1)) + 3
*2**(3/4)*sqrt((2*x**4 + 2*x**2 + 1)/(sqrt(2)*x**2 + 1)**2)*(2 + 3*sqrt(2))*(sqr
t(2)*x**2 + 1)*elliptic_pi(-11*sqrt(2)/24 + 1/2, 2*atan(2**(1/4)*x), -sqrt(2)/4
+ 1/2)/(32*(-3*sqrt(2) + 2)*sqrt(2*x**4 + 2*x**2 + 1)) + 3*sqrt(15)*atan(sqrt(15
)*x/(3*sqrt(2*x**4 + 2*x**2 + 1)))/40

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Mathematica [C]  time = 0.142511, size = 127, normalized size = 0.32 \[ -\frac{\sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \left (-(1+4 i) F\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+(1+i) E\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+3 i \Pi \left (\frac{1}{3}+\frac{i}{3};\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )\right )}{4 \sqrt{1-i} \sqrt{2 x^4+2 x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*((1 + I)*EllipticE[I*ArcSinh[Sqrt[
1 - I]*x], I] - (1 + 4*I)*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + (3*I)*Ellipti
cPi[1/3 + I/3, I*ArcSinh[Sqrt[1 - I]*x], I]))/(4*Sqrt[1 - I]*Sqrt[1 + 2*x^2 + 2*
x^4])

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Maple [C]  time = 0.03, size = 222, normalized size = 0.6 \[ -{\frac{3\,{\it EllipticF} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{4\,\sqrt{-1+i}}\sqrt{1+ \left ( 1-i \right ){x}^{2}}\sqrt{1+ \left ( 1+i \right ){x}^{2}}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{ \left ({\frac{1}{4}}-{\frac{i}{4}} \right ) \left ({\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) -{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) \right ) }{\sqrt{-1+i}}\sqrt{1+ \left ( 1-i \right ){x}^{2}}\sqrt{1+ \left ( 1+i \right ){x}^{2}}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{3}{4\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\it EllipticPi} \left ( x\sqrt{-1+i},{\frac{1}{3}}+{\frac{i}{3}},{\frac{\sqrt{-1-i}}{\sqrt{-1+i}}} \right ){\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(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. \[{\rm integral}\left (\frac{x^{4}}{\sqrt{2 \, x^{4} + 2 \, x^{2} + 1}{\left (2 \, x^{2} + 3\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/(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. \[ \int \frac{x^{4}}{\left (2 x^{2} + 3\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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