3.323 \(\int \frac{\left (1+2 x^2+2 x^4\right )^{3/2}}{x^6 \left (3-2 x^2\right )} \, dx\)

Optimal. Leaf size=665 \[ \frac{262 \sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{135 \left (\sqrt{2} x^2+1\right )}-\frac{262 \sqrt{2 x^4+2 x^2+1}}{135 x}+\frac{17}{27} \sqrt{\frac{17}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )+\frac{2^{3/4} \left (37+23 \sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{135 \sqrt{2 x^4+2 x^2+1}}+\frac{289 \sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{27 \left (2+3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{17 \left (3-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{27 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{262 \sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{135 \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left (3-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12+11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{81\ 2^{3/4} \left (2+3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\left (40 x^2+3\right ) \sqrt{2 x^4+2 x^2+1}}{45 x^5}+\frac{74 \sqrt{2 x^4+2 x^2+1}}{135 x^3} \]

[Out]

(74*Sqrt[1 + 2*x^2 + 2*x^4])/(135*x^3) - (262*Sqrt[1 + 2*x^2 + 2*x^4])/(135*x) -
((3 + 40*x^2)*Sqrt[1 + 2*x^2 + 2*x^4])/(45*x^5) + (262*Sqrt[2]*x*Sqrt[1 + 2*x^2
+ 2*x^4])/(135*(1 + Sqrt[2]*x^2)) + (17*Sqrt[17/3]*ArcTanh[(Sqrt[17/3]*x)/Sqrt[
1 + 2*x^2 + 2*x^4]])/27 - (262*2^(1/4)*(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])/(135*Sqr
t[1 + 2*x^2 + 2*x^4]) - (17*(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])/(27*2
^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4]) + (289*2^(1/4)*(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])/
(27*(2 + 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4]) + (2^(3/4)*(37 + 23*Sqrt[2])*(1 + S
qrt[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])/(135*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 + 1
1*Sqrt[2])/24, 2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(81*2^(3/4)*(2 + 3*Sqrt[2]
)*Sqrt[1 + 2*x^2 + 2*x^4])

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Rubi [A]  time = 0.957133, antiderivative size = 665, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276 \[ \frac{262 \sqrt{2} \sqrt{2 x^4+2 x^2+1} x}{135 \left (\sqrt{2} x^2+1\right )}-\frac{262 \sqrt{2 x^4+2 x^2+1}}{135 x}+\frac{17}{27} \sqrt{\frac{17}{3}} \tanh ^{-1}\left (\frac{\sqrt{\frac{17}{3}} x}{\sqrt{2 x^4+2 x^2+1}}\right )+\frac{2^{3/4} \left (37+23 \sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{135 \sqrt{2 x^4+2 x^2+1}}+\frac{289 \sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{27 \left (2+3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{17 \left (3-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{27 \sqrt [4]{2} \sqrt{2 x^4+2 x^2+1}}-\frac{262 \sqrt [4]{2} \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{135 \sqrt{2 x^4+2 x^2+1}}-\frac{289 \left (3-\sqrt{2}\right ) \left (\sqrt{2} x^2+1\right ) \sqrt{\frac{2 x^4+2 x^2+1}{\left (\sqrt{2} x^2+1\right )^2}} \Pi \left (\frac{1}{24} \left (12+11 \sqrt{2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac{1}{4} \left (2-\sqrt{2}\right )\right )}{81\ 2^{3/4} \left (2+3 \sqrt{2}\right ) \sqrt{2 x^4+2 x^2+1}}-\frac{\left (40 x^2+3\right ) \sqrt{2 x^4+2 x^2+1}}{45 x^5}+\frac{74 \sqrt{2 x^4+2 x^2+1}}{135 x^3} \]

Warning: Unable to verify antiderivative.

[In]  Int[(1 + 2*x^2 + 2*x^4)^(3/2)/(x^6*(3 - 2*x^2)),x]

[Out]

(74*Sqrt[1 + 2*x^2 + 2*x^4])/(135*x^3) - (262*Sqrt[1 + 2*x^2 + 2*x^4])/(135*x) -
((3 + 40*x^2)*Sqrt[1 + 2*x^2 + 2*x^4])/(45*x^5) + (262*Sqrt[2]*x*Sqrt[1 + 2*x^2
+ 2*x^4])/(135*(1 + Sqrt[2]*x^2)) + (17*Sqrt[17/3]*ArcTanh[(Sqrt[17/3]*x)/Sqrt[
1 + 2*x^2 + 2*x^4]])/27 - (262*2^(1/4)*(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])/(135*Sqr
t[1 + 2*x^2 + 2*x^4]) - (17*(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])/(27*2
^(1/4)*Sqrt[1 + 2*x^2 + 2*x^4]) + (289*2^(1/4)*(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])/
(27*(2 + 3*Sqrt[2])*Sqrt[1 + 2*x^2 + 2*x^4]) + (2^(3/4)*(37 + 23*Sqrt[2])*(1 + S
qrt[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])/(135*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 + 1
1*Sqrt[2])/24, 2*ArcTan[2^(1/4)*x], (2 - Sqrt[2])/4])/(81*2^(3/4)*(2 + 3*Sqrt[2]
)*Sqrt[1 + 2*x^2 + 2*x^4])

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Rubi in Sympy [A]  time = 86.9038, size = 597, normalized size = 0.9 \[ \frac{262 \sqrt{2} x \sqrt{2 x^{4} + 2 x^{2} + 1}}{135 \left (\sqrt{2} x^{2} + 1\right )} - \frac{262 \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 )}{135 \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{17 \sqrt [4]{2} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (- 6 \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 )}{108 \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 (184 + 148 \sqrt{2}\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 )}{540 \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{289 \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 )}{27 \left (2 + 3 \sqrt{2}\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{289 \cdot 2^{\frac{3}{4}} \sqrt{\frac{2 x^{4} + 2 x^{2} + 1}{\left (\sqrt{2} x^{2} + 1\right )^{2}}} \left (- 3 \sqrt{2} + 2\right ) \left (\sqrt{2} x^{2} + 1\right ) \Pi \left (\frac{1}{2} + \frac{11 \sqrt{2}}{24}; 2 \operatorname{atan}{\left (\sqrt [4]{2} x \right )}\middle | - \frac{\sqrt{2}}{4} + \frac{1}{2}\right )}{324 \left (2 + 3 \sqrt{2}\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}} + \frac{17 \sqrt{51} \operatorname{atanh}{\left (\frac{\sqrt{51} x}{3 \sqrt{2 x^{4} + 2 x^{2} + 1}} \right )}}{81} - \frac{262 \sqrt{2 x^{4} + 2 x^{2} + 1}}{135 x} + \frac{74 \sqrt{2 x^{4} + 2 x^{2} + 1}}{135 x^{3}} - \frac{\left (40 x^{2} + 3\right ) \sqrt{2 x^{4} + 2 x^{2} + 1}}{45 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**4+2*x**2+1)**(3/2)/x**6/(-2*x**2+3),x)

[Out]

262*sqrt(2)*x*sqrt(2*x**4 + 2*x**2 + 1)/(135*(sqrt(2)*x**2 + 1)) - 262*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)/(135*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)*(-6*sqrt(2) + 4)*(sqrt(2)*
x**2 + 1)*elliptic_f(2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(108*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)*(184 + 1
48*sqrt(2))*(sqrt(2)*x**2 + 1)*elliptic_f(2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/
(540*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)/(27*(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)*ellipti
c_pi(1/2 + 11*sqrt(2)/24, 2*atan(2**(1/4)*x), -sqrt(2)/4 + 1/2)/(324*(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)))/81 - 262*sqrt(2*x**4 + 2*x**2 + 1)/(135*x) + 74*sqrt(2*x**4 + 2*x*
*2 + 1)/(135*x**3) - (40*x**2 + 3)*sqrt(2*x**4 + 2*x**2 + 1)/(45*x**5)

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Mathematica [C]  time = 0.245698, size = 224, normalized size = 0.34 \[ -\frac{1572 x^8+1848 x^6+1116 x^4+192 x^2+(543-1329 i) \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^5 F\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+786 i \sqrt{1-i} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^5 E\left (\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )-1445 (1-i)^{3/2} \sqrt{1+(1-i) x^2} \sqrt{1+(1+i) x^2} x^5 \Pi \left (-\frac{1}{3}-\frac{i}{3};\left .i \sinh ^{-1}\left (\sqrt{1-i} x\right )\right |i\right )+27}{405 x^5 \sqrt{2 x^4+2 x^2+1}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + 2*x^2 + 2*x^4)^(3/2)/(x^6*(3 - 2*x^2)),x]

[Out]

-(27 + 192*x^2 + 1116*x^4 + 1848*x^6 + 1572*x^8 + (786*I)*Sqrt[1 - I]*x^5*Sqrt[1
+ (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*EllipticE[I*ArcSinh[Sqrt[1 - I]*x], I] + (
543 - 1329*I)*Sqrt[1 - I]*x^5*Sqrt[1 + (1 - I)*x^2]*Sqrt[1 + (1 + I)*x^2]*Ellipt
icF[I*ArcSinh[Sqrt[1 - I]*x], I] - 1445*(1 - I)^(3/2)*x^5*Sqrt[1 + (1 - I)*x^2]*
Sqrt[1 + (1 + I)*x^2]*EllipticPi[-1/3 - I/3, I*ArcSinh[Sqrt[1 - I]*x], I])/(405*
x^5*Sqrt[1 + 2*x^2 + 2*x^4])

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Maple [C]  time = 0.027, size = 549, normalized size = 0.8 \[ -{\frac{1}{15\,{x}^{5}}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}-{\frac{46}{135\,{x}^{3}}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}-{\frac{262}{135\,x}\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}+{\frac{184\,{\it EllipticF} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{45\,\sqrt{-1+i}}\sqrt{1+ \left ( 1-i \right ){x}^{2}}\sqrt{1+ \left ( 1+i \right ){x}^{2}}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{ \left ({\frac{52}{15}}-{\frac{52\,i}{15}} \right ) \left ({\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) -{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) \right ) }{\sqrt{-1+i}}\sqrt{1+ \left ( 1-i \right ){x}^{2}}\sqrt{1+ \left ( 1+i \right ){x}^{2}}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{236\,{\it EllipticF} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{45\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{{\frac{206\,i}{135}}{\it EllipticF} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}-{\frac{206\,{\it EllipticE} \left ( x\sqrt{-1+i},1/2\,\sqrt{2}+i/2\sqrt{2} \right ) }{135\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{{\frac{206\,i}{135}}{\it EllipticE} \left ( x\sqrt{-1+i},{\frac{\sqrt{2}}{2}}+{\frac{i}{2}}\sqrt{2} \right ) }{\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}}+{\frac{578}{81\,\sqrt{-1+i}}\sqrt{-i{x}^{2}+{x}^{2}+1}\sqrt{i{x}^{2}+{x}^{2}+1}{\it EllipticPi} \left ( x\sqrt{-1+i},-{\frac{1}{3}}-{\frac{i}{3}},{\frac{\sqrt{-1-i}}{\sqrt{-1+i}}} \right ){\frac{1}{\sqrt{2\,{x}^{4}+2\,{x}^{2}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^4+2*x^2+1)^(3/2)/x^6/(-2*x^2+3),x)

[Out]

-1/15*(2*x^4+2*x^2+1)^(1/2)/x^5-46/135*(2*x^4+2*x^2+1)^(1/2)/x^3-262/135*(2*x^4+
2*x^2+1)^(1/2)/x+184/45/(-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))+(-52/15+5
2/15*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)))-236/45/(-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))-206/135*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))-206/135/(-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))+206/135*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))+578/81/(-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. \[ -\int \frac{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (2 \, x^{2} - 3\right )} x^{6}}\,{d x} \]

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^6),x, algorithm="maxima")

[Out]

-integrate((2*x^4 + 2*x^2 + 1)^(3/2)/((2*x^2 - 3)*x^6), x)

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

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^6),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{2 x^{4} + 2 x^{2} + 1}}{2 x^{8} - 3 x^{6}}\, dx - \int \frac{2 x^{2} \sqrt{2 x^{4} + 2 x^{2} + 1}}{2 x^{8} - 3 x^{6}}\, dx - \int \frac{2 x^{4} \sqrt{2 x^{4} + 2 x^{2} + 1}}{2 x^{8} - 3 x^{6}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**4+2*x**2+1)**(3/2)/x**6/(-2*x**2+3),x)

[Out]

-Integral(sqrt(2*x**4 + 2*x**2 + 1)/(2*x**8 - 3*x**6), x) - Integral(2*x**2*sqrt
(2*x**4 + 2*x**2 + 1)/(2*x**8 - 3*x**6), x) - Integral(2*x**4*sqrt(2*x**4 + 2*x*
*2 + 1)/(2*x**8 - 3*x**6), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{{\left (2 \, x^{4} + 2 \, x^{2} + 1\right )}^{\frac{3}{2}}}{{\left (2 \, x^{2} - 3\right )} x^{6}}\,{d x} \]

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^6),x, algorithm="giac")

[Out]

integrate(-(2*x^4 + 2*x^2 + 1)^(3/2)/((2*x^2 - 3)*x^6), x)