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

Optimal. Leaf size=145 \[ \frac{1}{5} \left (x^2+2\right ) \sqrt{x^4+x^2+1} x+\frac{3 \sqrt{x^4+x^2+1} x}{5 \left (x^2+1\right )}+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{5 \sqrt{x^4+x^2+1}}-\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{5 \sqrt{x^4+x^2+1}} \]

[Out]

(3*x*Sqrt[1 + x^2 + x^4])/(5*(1 + x^2)) + (x*(2 + x^2)*Sqrt[1 + x^2 + x^4])/5 -
(3*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/4])/(5*S
qrt[1 + x^2 + x^4]) + (3*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticF[2
*ArcTan[x], 1/4])/(5*Sqrt[1 + x^2 + x^4])

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Rubi [A]  time = 0.103504, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{1}{5} \left (x^2+2\right ) \sqrt{x^4+x^2+1} x+\frac{3 \sqrt{x^4+x^2+1} x}{5 \left (x^2+1\right )}+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{5 \sqrt{x^4+x^2+1}}-\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{5 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(3*x*Sqrt[1 + x^2 + x^4])/(5*(1 + x^2)) + (x*(2 + x^2)*Sqrt[1 + x^2 + x^4])/5 -
(3*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/4])/(5*S
qrt[1 + x^2 + x^4]) + (3*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticF[2
*ArcTan[x], 1/4])/(5*Sqrt[1 + x^2 + x^4])

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Rubi in Sympy [A]  time = 18.4321, size = 138, normalized size = 0.95 \[ \frac{x \left (3 x^{2} + 6\right ) \sqrt{x^{4} + x^{2} + 1}}{15} + \frac{3 x \sqrt{x^{4} + x^{2} + 1}}{5 \left (x^{2} + 1\right )} - \frac{3 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{5 \sqrt{x^{4} + x^{2} + 1}} + \frac{3 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{5 \sqrt{x^{4} + x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(3*x**2 + 6)*sqrt(x**4 + x**2 + 1)/15 + 3*x*sqrt(x**4 + x**2 + 1)/(5*(x**2 + 1
)) - 3*sqrt((x**4 + x**2 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic_e(2*atan(x), 1/
4)/(5*sqrt(x**4 + x**2 + 1)) + 3*sqrt((x**4 + x**2 + 1)/(x**2 + 1)**2)*(x**2 + 1
)*elliptic_f(2*atan(x), 1/4)/(5*sqrt(x**4 + x**2 + 1))

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Mathematica [C]  time = 0.327481, size = 168, normalized size = 1.16 \[ \frac{x^7+3 x^5+3 x^3+\frac{3}{2} \sqrt{2+\left (1-i \sqrt{3}\right ) x^2} \sqrt{2+\left (1+i \sqrt{3}\right ) x^2} F\left (\sin ^{-1}\left (\frac{1}{2} \left (i \sqrt{3} x+x\right )\right )|\frac{1}{2} i \left (i+\sqrt{3}\right )\right )+3 \sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+2 x}{5 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(2*x + 3*x^3 + 3*x^5 + x^7 + 3*(-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)] + (3*Sqrt[2 + (1 - I*
Sqrt[3])*x^2]*Sqrt[2 + (1 + I*Sqrt[3])*x^2]*EllipticF[ArcSin[(x + I*Sqrt[3]*x)/2
], (I/2)*(I + Sqrt[3])])/2)/(5*Sqrt[1 + x^2 + x^4])

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Maple [C]  time = 0.008, size = 233, normalized size = 1.6 \[{\frac{2\,x}{5}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{6}{5\,\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}-{\frac{12}{5\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}+{\frac{{x}^{3}}{5}\sqrt{{x}^{4}+{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*x*(x^4+x^2+1)^(1/2)+6/5/(-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))-12/5/(-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)))+1/5*x^3*
(x^4+x^2+1)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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