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

Optimal. Leaf size=463 \[ -\frac{1}{14} \left (2 x^4+2 x^2+1\right )^{3/2} x-\frac{2211 \sqrt{2 x^4+2 x^2+1} x}{140 \sqrt{2} \left (\sqrt{2} x^2+1\right )}-\frac{213}{140} \sqrt{2 x^4+2 x^2+1} x+\frac{17}{16} \sqrt{51} \tanh ^{-1}\left (\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )-\frac{3 \left (514+2717 \sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{140\ 2^{3/4} \left (2+3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}+\frac{2211 \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{140\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left (3-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12+11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{16\ 2^{3/4} \left (2+3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{27}{70} \sqrt{2 x^4+2 x^2+1} x^3 \]

[Out]

(-213*x*Sqrt[1 + 2*x^2 + 2*x^4])/140 - (27*x^3*Sqrt[1 + 2*x^2 + 2*x^4])/70 - (22
11*x*Sqrt[1 + 2*x^2 + 2*x^4])/(140*Sqrt[2]*(1 + Sqrt[2]*x^2)) - (x*(1 + 2*x^2 +
2*x^4)^(3/2))/14 + (17*Sqrt[51]*ArcTanh[(Sqrt[17/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])
/16 + (2211*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2]*Elli
pticE[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(140*2^(3/4)*Sqrt[1 + 2*x^2 + 2*x^4
]) - (3*(514 + 2717*Sqrt[2])*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqr
t[2]*x^2)^2]*EllipticF[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(140*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*ArcT
an[2^(1/4)*x], (2 - Sqrt[2])/4])/(16*2^(3/4)*(2 + 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*
x^4])

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Rubi [A]  time = 1.1918, antiderivative size = 888, normalized size of antiderivative = 1.92, number of steps used = 18, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276 \[ -\frac{1}{14} x \left (2 x^4+2 x^2+1\right )^{3/2}-\frac{3}{35} x \left (x^2+2\right ) \sqrt{2 x^4+2 x^2+1}-\frac{3}{20} x \left (2 x^2+9\right ) \sqrt{2 x^4+2 x^2+1}-\frac{6 \sqrt{2} x \sqrt{2 x^4+2 x^2+1}}{35 \left (\sqrt{2} x^2+1\right )}-\frac{309 x \sqrt{2 x^4+2 x^2+1}}{20 \sqrt{2} \left (\sqrt{2} x^2+1\right )}+\frac{17}{16} \sqrt{51} \tanh ^{-1}\left (\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )+\frac{6 \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}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{35 \sqrt{2 x^4+2 x^2+1}}+\frac{309 \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 )}{20\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}-\frac{3 \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 )}{20\ 2^{3/4} \sqrt{2 x^4+2 x^2+1}}+\frac{867 \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 (2+3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{3 \left (3+2 \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 )}{70 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{51 \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 )}{16 \sqrt [4]{2} \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 )}{16\ 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^2*(1 + 2*x^2 + 2*x^4)^(3/2))/(3 - 2*x^2),x]

[Out]

(-3*x*(2 + x^2)*Sqrt[1 + 2*x^2 + 2*x^4])/35 - (3*x*(9 + 2*x^2)*Sqrt[1 + 2*x^2 +
2*x^4])/20 - (309*x*Sqrt[1 + 2*x^2 + 2*x^4])/(20*Sqrt[2]*(1 + Sqrt[2]*x^2)) - (6
*Sqrt[2]*x*Sqrt[1 + 2*x^2 + 2*x^4])/(35*(1 + Sqrt[2]*x^2)) - (x*(1 + 2*x^2 + 2*x
^4)^(3/2))/14 + (17*Sqrt[51]*ArcTanh[(Sqrt[17/3]*x)/Sqrt[1 + 2*x^2 + 2*x^4]])/16
+ (309*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2]*Elliptic
E[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(20*2^(3/4)*Sqrt[1 + 2*x^2 + 2*x^4]) +
(6*2^(1/4)*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2]*Ellip
ticE[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(35*Sqrt[1 + 2*x^2 + 2*x^4]) - (51*(
5 + Sqrt[2])*(1 + Sqrt[2]*x^2)*Sqrt[(1 + 2*x^2 + 2*x^4)/(1 + Sqrt[2]*x^2)^2]*Ell
ipticF[2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(16*2^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4
]) - (3*(3 + 2*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])/(70*2^(1/4)*Sqrt[1 + 2*
x^2 + 2*x^4]) + (867*(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^(3/4)*(2 + 3*Sqrt[2])
*Sqrt[1 + 2*x^2 + 2*x^4]) - (3*(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])/
(20*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*Arc
Tan[2^(1/4)*x], (2 - Sqrt[2])/4])/(16*2^(3/4)*(2 + 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2
*x^4])

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Rubi in Sympy [A]  time = 134.069, size = 796, normalized size = 1.72 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(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)/20 - x*(12*x**2 + 24)*sqrt(2*x**4 + 2*
x**2 + 1)/140 - x*(2*x**4 + 2*x**2 + 1)**(3/2)/14 - 5*x*sqrt(2*x**4 + 2*x**2 + 1
)/4 - 2211*sqrt(2)*x*sqrt(2*x**4 + 2*x**2 + 1)/(280*(sqrt(2)*x**2 + 1)) + 2211*2
**(1/4)*sqrt((2*x**4 + 2*x**2 + 1)/(sqrt(2)*x**2 + 1)**2)*(sqrt(2)*x**2 + 1)*ell
iptic_e(2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(280*sqrt(2*x**4 + 2*x**2 + 1)) -
51*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)/(64*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*sqrt(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)*(48 + 36*sqrt(2))*(sqrt(2)*x**2 + 1)*elliptic_f(2*atan(2**(1/4)
*x), -sqrt(2)/4 + 1/2)/(560*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)/(80*sqrt(2*x**4 + 2*x**2 + 1)) + 867*2*
*(1/4)*sqrt((2*x**4 + 2*x**2 + 1)/(sqrt(2)*x**2 + 1)**2)*(sqrt(2)*x**2 + 1)*elli
ptic_f(2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(16*(2 + 3*sqrt(2))*sqrt(2*x**4 + 2
*x**2 + 1)) + 289*2**(3/4)*sqrt((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)/(64*(2 + 3*sqrt(2))*sqrt(2*x**4 + 2*x**2 + 1)) + 17*sqrt
(51)*atanh(sqrt(51)*x/(3*sqrt(2*x**4 + 2*x**2 + 1)))/16

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Mathematica [C]  time = 0.213384, size = 214, normalized size = 0.46 \[ \frac{-160 x^9-752 x^7-2456 x^5-2080 x^3-(9669-5247 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} F\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+4422 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} E\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+10115 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} \Pi \left (-\frac{1}{3}-\frac{i}{3};\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )-892 x}{560 \sqrt{2 x^4+2 x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(-892*x - 2080*x^3 - 2456*x^5 - 752*x^7 - 160*x^9 + (4422*I)*Sqrt[1 - I]*Sqrt[1
+ (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] - (9
669 - 5247*I)*Sqrt[1 - I]*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticF[
I*ArcSinh[Sqrt[1 - I]*x], I] + 10115*(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])/(560*Sqrt[1
+ 2*x^2 + 2*x^4])

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Maple [C]  time = 0.032, size = 547, normalized size = 1.2 \[ -{\frac{{x}^{5}}{7}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}-{\frac{37\,{x}^{3}}{70}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}-{\frac{223\,x}{140}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}-{\frac{9\,{\it EllipticF} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{35\,\sqrt{-1+i}}\sqrt{1+ \left ( 1-i \right ){x}^{2}}\sqrt{1+ \left ( 1+i \right ){x}^{2}}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{ \left ({\frac{6}{35}}-{\frac{6\,i}{35}} \right ) \left ({\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) -{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) \right ) }{\sqrt{-1+i}}\sqrt{1+ \left ( 1-i \right ){x}^{2}}\sqrt{1+ \left ( 1+i \right ){x}^{2}}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{531\,{\it EllipticF} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{20\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{{\frac{309\,i}{40}}{\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{309\,{\it EllipticE} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{40\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{{\frac{309\,i}{40}}{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{289}{8\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\it EllipticPi} \left ( x\sqrt{-1+i},-{\frac{1}{3}}-{\frac{i}{3}},{\frac{\sqrt{-1-i}}{\sqrt{-1+i}}} \right ){\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/7*x^5*(2*x^4+2*x^2+1)^(1/2)-37/70*x^3*(2*x^4+2*x^2+1)^(1/2)-223/140*x*(2*x^4+
2*x^2+1)^(1/2)-9/35/(-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))+(6/35-6/35*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)*(Ell
ipticF(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)))-531/20/(-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)*EllipticF(x*(-1+I)^(1/2),1/2*2^(1/2)+1/2*I*2^(1/2))-309
/40*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))-309/40/(-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)*EllipticE(x*(-1+I)^(1/2
),1/2*2^(1/2)+1/2*I*2^(1/2))+309/40*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))+289/8/(-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. \[ -\int \frac{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac{3}{2}} x^{2}}{2 \, x^{2} - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-(2*x^6 + 2*x^4 + x^2)*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. \[ - \int \frac{x^{2} \sqrt{2 x^{4} + 2 x^{2} + 1}}{2 x^{2} - 3}\, dx - \int \frac{2 x^{4} \sqrt{2 x^{4} + 2 x^{2} + 1}}{2 x^{2} - 3}\, dx - \int \frac{2 x^{6} \sqrt{2 x^{4} + 2 x^{2} + 1}}{2 x^{2} - 3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(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) - Integral(2*x**6*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. \[ \int -\frac{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac{3}{2}} x^{2}}{2 \, x^{2} - 3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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