3.340 \(\int \frac{x^8}{\left (3+2 x^2\right ) \left (1+2 x^2+2 x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=449 \[ \frac{\sqrt{2 x^4+2 x^2+1} x}{10 \sqrt{2} \left (\sqrt{2} x^2+1\right )}+\frac{1}{20} \sqrt{2 x^4+2 x^2+1} x+\frac{27}{80} \sqrt{\frac{3}{5}} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )+\frac{\left (7 \sqrt{2}-2\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{8\ 2^{3/4} \left (3 \sqrt{2}-2\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{10\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{27 \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{80\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}+\frac{\left (1-2 x^2\right ) x^3}{20 \sqrt{2 x^4+2 x^2+1}} \]

[Out]

(x^3*(1 - 2*x^2))/(20*Sqrt[1 + 2*x^2 + 2*x^4]) + (x*Sqrt[1 + 2*x^2 + 2*x^4])/20
+ (x*Sqrt[1 + 2*x^2 + 2*x^4])/(10*Sqrt[2]*(1 + Sqrt[2]*x^2)) + (27*Sqrt[3/5]*Arc
Tan[(Sqrt[5/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/80 - ((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
])/(10*2^(3/4)*Sqrt[1 + 2*x^2 + 2*x^4]) + ((-2 + 7*Sqrt[2])*(1 + Sqrt[2]*x^2)*Sq
rt[(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^(3/4)*(-2 + 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4]) + (27*(3 + Sqr
t[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])/(80*2^(3/4)*(2 - 3
*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4])

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Rubi [A]  time = 0.99187, antiderivative size = 636, normalized size of antiderivative = 1.42, number of steps used = 13, number of rules used = 9, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.31 \[ \frac{\sqrt{2 x^4+2 x^2+1} x}{10 \sqrt{2} \left (\sqrt{2} x^2+1\right )}+\frac{1}{20} \sqrt{2 x^4+2 x^2+1} x+\frac{27}{80} \sqrt{\frac{3}{5}} \tan ^{-1}\left (\frac{\sqrt{\frac{5}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )-\frac{\left (7+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{40\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{9 \left (3-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{80 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{81 \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{40\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{10\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{27 \left (3+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12-11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{80\ 2^{3/4} \left (2-3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}+\frac{\left (1-2 x^2\right ) x^3}{20 \sqrt{2 x^4+2 x^2+1}} \]

Warning: Unable to verify antiderivative.

[In]  Int[x^8/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)),x]

[Out]

(x^3*(1 - 2*x^2))/(20*Sqrt[1 + 2*x^2 + 2*x^4]) + (x*Sqrt[1 + 2*x^2 + 2*x^4])/20
+ (x*Sqrt[1 + 2*x^2 + 2*x^4])/(10*Sqrt[2]*(1 + Sqrt[2]*x^2)) + (27*Sqrt[3/5]*Arc
Tan[(Sqrt[5/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/80 - ((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
])/(10*2^(3/4)*Sqrt[1 + 2*x^2 + 2*x^4]) - (81*(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])/(
40*2^(3/4)*(2 - 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4]) - (9*(3 - 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])/(80*2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4]) - ((7 + 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])/(40*2^(3/4)*Sqrt[1 + 2*x^2 + 2*x^4]) + (27*(3 + S
qrt[2])*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2]*Elliptic
Pi[(12 - 11*Sqrt[2])/24, 2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(80*2^(3/4)*(2 -
3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4])

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Rubi in Sympy [A]  time = 84.5409, size = 643, normalized size = 1.43 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(2*x**2+3)/(2*x**4+2*x**2+1)**(3/2),x)

[Out]

x**3*(-4*x**2 + 2)/(40*sqrt(2*x**4 + 2*x**2 + 1)) + x*sqrt(2*x**4 + 2*x**2 + 1)/
20 + sqrt(2)*x*sqrt(2*x**4 + 2*x**2 + 1)/(20*(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*at
an(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(20*sqrt(2*x**4 + 2*x**2 + 1)) - 27*2**(3/4)*s
qrt((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)/(160*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)*(12*sqrt(2) + 84)*(sqrt(2)*x**
2 + 1)*elliptic_f(2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(960*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)/(80*sqrt(2*x**4 + 2*x
**2 + 1)) - 81*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)/(80*(-3*sqrt(2) +
2)*sqrt(2*x**4 + 2*x**2 + 1)) + 27*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)/(320*(-3*sqrt(2) + 2)*sqrt(2*x**4 + 2*x
**2 + 1)) + 27*sqrt(15)*atan(sqrt(15)*x/(3*sqrt(2*x**4 + 2*x**2 + 1)))/400

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Mathematica [C]  time = 0.203012, size = 199, normalized size = 0.44 \[ \frac{12 x^3-(29-33 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )-4 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+27 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left (\frac{1}{3}+\frac{i}{3};\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+4 x}{80 \sqrt{2 x^4+2 x^2+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/((3 + 2*x^2)*(1 + 2*x^2 + 2*x^4)^(3/2)),x]

[Out]

(4*x + 12*x^3 - (4*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*El
lipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (29 - 33*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*
x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[1 - I]*x], I] + 27*(1 - I)^(
3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticPi[1/3 + I/3, I*ArcSinh
[Sqrt[1 - I]*x], I])/(80*Sqrt[1 + 2*x^2 + 2*x^4])

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Maple [C]  time = 0.052, size = 603, normalized size = 1.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(2*x^2+3)/(2*x^4+2*x^2+1)^(3/2),x)

[Out]

27/16*x^3/(2*x^4+2*x^2+1)^(1/2)-11/4/(-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))+(47/32-47/32*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)))-9/2*(-1/4*x^3-1/8*x)/(2*x^4+2*x^2+1)^(1
/2)+3*(1/8*x^3+1/8*x)/(2*x^4+2*x^2+1)^(1/2)+1/8*x/(2*x^4+2*x^2+1)^(1/2)-81/4*(3/
20*x^3+1/20*x)/(2*x^4+2*x^2+1)^(1/2)+81/160/(-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))+243/160*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))+243/160/(-
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)*Ellipt
icE(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))-243/160*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))+27/40/(-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. \[ \int \frac{x^{8}}{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac{3}{2}}{\left (2 \, x^{2} + 3\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((2*x^4 + 2*x^2 + 1)^(3/2)*(2*x^2 + 3)),x, algorithm="maxima")

[Out]

integrate(x^8/((2*x^4 + 2*x^2 + 1)^(3/2)*(2*x^2 + 3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{8}}{{\left (4 \, x^{6} + 10 \, x^{4} + 8 \, x^{2} + 3\right )} \sqrt{2 \, x^{4} + 2 \, x^{2} + 1}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((2*x^4 + 2*x^2 + 1)^(3/2)*(2*x^2 + 3)),x, algorithm="fricas")

[Out]

integral(x^8/((4*x^6 + 10*x^4 + 8*x^2 + 3)*sqrt(2*x^4 + 2*x^2 + 1)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\left (2 x^{2} + 3\right ) \left (2 x^{4} + 2 x^{2} + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(2*x**2+3)/(2*x**4+2*x**2+1)**(3/2),x)

[Out]

Integral(x**8/((2*x**2 + 3)*(2*x**4 + 2*x**2 + 1)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac{3}{2}}{\left (2 \, x^{2} + 3\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((2*x^4 + 2*x^2 + 1)^(3/2)*(2*x^2 + 3)),x, algorithm="giac")

[Out]

integrate(x^8/((2*x^4 + 2*x^2 + 1)^(3/2)*(2*x^2 + 3)), x)