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

Optimal. Leaf size=164 \[ \frac{1}{7} x \left (x^4+x^2+1\right )^{3/2}+\frac{2}{21} x \left (3 x^2+4\right ) \sqrt{x^4+x^2+1}+\frac{2 x \sqrt{x^4+x^2+1}}{3 \left (x^2+1\right )}+\frac{4 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{7 \sqrt{x^4+x^2+1}}-\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]

[Out]

(2*x*Sqrt[1 + x^2 + x^4])/(3*(1 + x^2)) + (2*x*(4 + 3*x^2)*Sqrt[1 + x^2 + x^4])/
21 + (x*(1 + x^2 + x^4)^(3/2))/7 - (2*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2
]*EllipticE[2*ArcTan[x], 1/4])/(3*Sqrt[1 + x^2 + x^4]) + (4*(1 + x^2)*Sqrt[(1 +
x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/4])/(7*Sqrt[1 + x^2 + x^4])

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Rubi [A]  time = 0.140456, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{1}{7} x \left (x^4+x^2+1\right )^{3/2}+\frac{2}{21} x \left (3 x^2+4\right ) \sqrt{x^4+x^2+1}+\frac{2 x \sqrt{x^4+x^2+1}}{3 \left (x^2+1\right )}+\frac{4 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{7 \sqrt{x^4+x^2+1}}-\frac{2 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{3 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x*Sqrt[1 + x^2 + x^4])/(3*(1 + x^2)) + (2*x*(4 + 3*x^2)*Sqrt[1 + x^2 + x^4])/
21 + (x*(1 + x^2 + x^4)^(3/2))/7 - (2*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2
]*EllipticE[2*ArcTan[x], 1/4])/(3*Sqrt[1 + x^2 + x^4]) + (4*(1 + x^2)*Sqrt[(1 +
x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x], 1/4])/(7*Sqrt[1 + x^2 + x^4])

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Rubi in Sympy [A]  time = 28.8372, size = 156, normalized size = 0.95 \[ \frac{x \left (\frac{30 x^{2}}{7} + \frac{40}{7}\right ) \sqrt{x^{4} + x^{2} + 1}}{15} + \frac{x \left (x^{4} + x^{2} + 1\right )^{\frac{3}{2}}}{7} + \frac{2 x \sqrt{x^{4} + x^{2} + 1}}{3 \left (x^{2} + 1\right )} - \frac{2 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{3 \sqrt{x^{4} + x^{2} + 1}} + \frac{4 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{7 \sqrt{x^{4} + x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(30*x**2/7 + 40/7)*sqrt(x**4 + x**2 + 1)/15 + x*(x**4 + x**2 + 1)**(3/2)/7 + 2
*x*sqrt(x**4 + x**2 + 1)/(3*(x**2 + 1)) - 2*sqrt((x**4 + x**2 + 1)/(x**2 + 1)**2
)*(x**2 + 1)*elliptic_e(2*atan(x), 1/4)/(3*sqrt(x**4 + x**2 + 1)) + 4*sqrt((x**4
+ x**2 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic_f(2*atan(x), 1/4)/(7*sqrt(x**4 +
x**2 + 1))

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Mathematica [C]  time = 0.232731, size = 162, normalized size = 0.99 \[ \frac{2 \sqrt [3]{-1} \left (5 \sqrt [3]{-1}-7\right ) \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+14 \sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+x \left (3 x^8+12 x^6+23 x^4+20 x^2+11\right )}{21 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(11 + 20*x^2 + 23*x^4 + 12*x^6 + 3*x^8) + 14*(-1)^(1/3)*Sqrt[1 + (-1)^(1/3)*x
^2]*Sqrt[1 - (-1)^(2/3)*x^2]*EllipticE[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)] + 2*
(-1)^(1/3)*(-7 + 5*(-1)^(1/3))*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]
*EllipticF[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(21*Sqrt[1 + x^2 + x^4])

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Maple [C]  time = 0.011, size = 248, normalized size = 1.5 \[{\frac{11\,x}{21}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{20}{21\,\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}-{\frac{8}{3\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}+{\frac{{x}^{5}}{7}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{3\,{x}^{3}}{7}\sqrt{{x}^{4}+{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

11/21*x*(x^4+x^2+1)^(1/2)+20/21/(-2+2*I*3^(1/2))^(1/2)*(1-(-1/2+1/2*I*3^(1/2))*x
^2)^(1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4+x^2+1)^(1/2)*EllipticF(1/2*x*(
-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2))-8/3/(-2+2*I*3^(1/2))^(1/2)*(1-
(-1/2+1/2*I*3^(1/2))*x^2)^(1/2)*(1-(-1/2-1/2*I*3^(1/2))*x^2)^(1/2)/(x^4+x^2+1)^(
1/2)/(I*3^(1/2)+1)*(EllipticF(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^
(1/2))-EllipticE(1/2*x*(-2+2*I*3^(1/2))^(1/2),1/2*(-2+2*I*3^(1/2))^(1/2)))+1/7*x
^5*(x^4+x^2+1)^(1/2)+3/7*x^3*(x^4+x^2+1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((x^4 + 2*x^2 + 1)*sqrt(x^4 + x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((x**2 - x + 1)*(x**2 + x + 1))*(x**2 + 1)**2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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