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

Optimal. Leaf size=428 \[ -\frac{1}{10} \left (2 x^2+9\right ) \sqrt{2 x^4+2 x^2+1} x-\frac{103 \sqrt{2 x^4+2 x^2+1} x}{10 \sqrt{2} \left (\sqrt{2} x^2+1\right )}+\frac{17}{8} \sqrt{\frac{17}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )-\frac{\left (66+383 \sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{10\ 2^{3/4} \left (2+3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}+\frac{103 \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{10\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left (3-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12+11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{24\ 2^{3/4} \left (2+3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}} \]

[Out]

-(x*(9 + 2*x^2)*Sqrt[1 + 2*x^2 + 2*x^4])/10 - (103*x*Sqrt[1 + 2*x^2 + 2*x^4])/(1
0*Sqrt[2]*(1 + Sqrt[2]*x^2)) + (17*Sqrt[17/3]*ArcTanh[(Sqrt[17/3]*x)/Sqrt[1 + 2*
x^2 + 2*x^4]])/8 + (103*(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]) - ((66 + 383*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])/(10*2^(3/4)
*(2 + 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4]) - (289*(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])/(24*2^(3/4)*(2 + 3*Sqrt[2])*Sqrt[1 + 2*x^
2 + 2*x^4])

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Rubi [A]  time = 0.653905, antiderivative size = 615, normalized size of antiderivative = 1.44, number of steps used = 11, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ -\frac{1}{10} \left (2 x^2+9\right ) \sqrt{2 x^4+2 x^2+1} x-\frac{103 \sqrt{2 x^4+2 x^2+1} x}{10 \sqrt{2} \left (\sqrt{2} x^2+1\right )}+\frac{17}{8} \sqrt{\frac{17}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )-\frac{\left (9+8 \sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{10\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{289 \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{4\ 2^{3/4} \left (2+3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{17 \left (5+\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{8 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}+\frac{103 \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{10\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left (3-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12+11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{24\ 2^{3/4} \left (2+3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(x*(9 + 2*x^2)*Sqrt[1 + 2*x^2 + 2*x^4])/10 - (103*x*Sqrt[1 + 2*x^2 + 2*x^4])/(1
0*Sqrt[2]*(1 + Sqrt[2]*x^2)) + (17*Sqrt[17/3]*ArcTanh[(Sqrt[17/3]*x)/Sqrt[1 + 2*
x^2 + 2*x^4]])/8 + (103*(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]) - (17*(5 + 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])/(8*2^(1/4)*Sq
rt[1 + 2*x^2 + 2*x^4]) + (289*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sq
rt[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]) - ((9 + 8*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])/(10*2^(3/4)*Sqrt[1 + 2*x^2 + 2*x^4]) - (289*(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])/(24*2^(3/4)*(2 + 3*Sqrt[2])*Sqrt[1 + 2*
x^2 + 2*x^4])

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Rubi in Sympy [A]  time = 97.2721, size = 663, normalized size = 1.55 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*(6*x**2 + 2)*sqrt(2*x**4 + 2*x**2 + 1)/30 - 5*x*sqrt(2*x**4 + 2*x**2 + 1)/6 -
103*sqrt(2)*x*sqrt(2*x**4 + 2*x**2 + 1)/(20*(sqrt(2)*x**2 + 1)) + 103*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)/(20*sqrt(2*x**4 + 2*x**2 + 1)) - 17*2**(1/
4)*sqrt((2*x**4 + 2*x**2 + 1)/(sqrt(2)*x**2 + 1)**2)*(4 + 10*sqrt(2))*(sqrt(2)*x
**2 + 1)*elliptic_f(2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(32*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)*(2 + 2*s
qrt(2))*(sqrt(2)*x**2 + 1)*elliptic_f(2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(24*
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)*elliptic_f(2*atan(2**(1/4)*x), -sqrt
(2)/4 + 1/2)/(120*sqrt(2*x**4 + 2*x**2 + 1)) + 289*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*(2 + 3*sqrt(2))*sqrt(2*x**4 + 2*x**2 + 1)) + 289*2**(3/4)*s
qrt((2*x**4 + 2*x**2 + 1)/(sqrt(2)*x**2 + 1)**2)*(-3*sqrt(2) + 2)*(sqrt(2)*x**2
+ 1)*elliptic_pi(1/2 + 11*sqrt(2)/24, 2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(96*
(2 + 3*sqrt(2))*sqrt(2*x**4 + 2*x**2 + 1)) + 17*sqrt(51)*atanh(sqrt(51)*x/(3*sqr
t(2*x**4 + 2*x**2 + 1)))/24

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Mathematica [C]  time = 0.198059, size = 209, normalized size = 0.49 \[ \frac{-48 x^7-264 x^5-240 x^3-(1371-753 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+618 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+1445 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left (-\frac{1}{3}-\frac{i}{3};\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )-108 x}{120 \sqrt{2 x^4+2 x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(-108*x - 240*x^3 - 264*x^5 - 48*x^7 + (618*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]
*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (1371 - 753*I)*S
qrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[I*ArcSinh[Sqrt[
1 - I]*x], I] + 1445*(1 - I)^(3/2)*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*E
llipticPi[-1/3 - I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(120*Sqrt[1 + 2*x^2 + 2*x^4]
)

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Maple [C]  time = 0.009, size = 377, normalized size = 0.9 \[ -{\frac{{x}^{3}}{5}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}-{\frac{9\,x}{10}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}-{\frac{177\,{\it EllipticF} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{10\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{{\frac{103\,i}{20}}{\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{103\,{\it EllipticE} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{20\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{{\frac{103\,i}{20}}{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{289}{12\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\it EllipticPi} \left ( x\sqrt{-1+i},-{\frac{1}{3}}-{\frac{i}{3}},{\frac{\sqrt{-1-i}}{\sqrt{-1+i}}} \right ){\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*x^3*(2*x^4+2*x^2+1)^(1/2)-9/10*x*(2*x^4+2*x^2+1)^(1/2)-177/10/(-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))-103/20*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))-103/20/(-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))+103/20*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)*Elliptic
E(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))+289/12/(-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. \[ -\int \frac{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac{3}{2}}}{2 \, x^{2} - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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