3.229 \(\int \left (1+x^2\right )^3 \sqrt{1+x^2+x^4} \, dx\)

Optimal. Leaf size=183 \[ \frac{1}{3} \left (x^4+x^2+1\right )^{3/2} x+\frac{2}{45} \left (6 x^2+7\right ) \sqrt{x^4+x^2+1} x+\frac{26 \sqrt{x^4+x^2+1} x}{45 \left (x^2+1\right )}+\frac{7 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{15 \sqrt{x^4+x^2+1}}-\frac{26 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{45 \sqrt{x^4+x^2+1}}+\frac{1}{9} \left (x^4+x^2+1\right )^{3/2} x^3 \]

[Out]

(26*x*Sqrt[1 + x^2 + x^4])/(45*(1 + x^2)) + (2*x*(7 + 6*x^2)*Sqrt[1 + x^2 + x^4]
)/45 + (x*(1 + x^2 + x^4)^(3/2))/3 + (x^3*(1 + x^2 + x^4)^(3/2))/9 - (26*(1 + x^
2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/4])/(45*Sqrt[1 + x
^2 + x^4]) + (7*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x
], 1/4])/(15*Sqrt[1 + x^2 + x^4])

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Rubi [A]  time = 0.179909, antiderivative size = 183, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{1}{3} \left (x^4+x^2+1\right )^{3/2} x+\frac{2}{45} \left (6 x^2+7\right ) \sqrt{x^4+x^2+1} x+\frac{26 \sqrt{x^4+x^2+1} x}{45 \left (x^2+1\right )}+\frac{7 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{15 \sqrt{x^4+x^2+1}}-\frac{26 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{45 \sqrt{x^4+x^2+1}}+\frac{1}{9} \left (x^4+x^2+1\right )^{3/2} x^3 \]

Antiderivative was successfully verified.

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

[Out]

(26*x*Sqrt[1 + x^2 + x^4])/(45*(1 + x^2)) + (2*x*(7 + 6*x^2)*Sqrt[1 + x^2 + x^4]
)/45 + (x*(1 + x^2 + x^4)^(3/2))/3 + (x^3*(1 + x^2 + x^4)^(3/2))/9 - (26*(1 + x^
2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticE[2*ArcTan[x], 1/4])/(45*Sqrt[1 + x
^2 + x^4]) + (7*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*EllipticF[2*ArcTan[x
], 1/4])/(15*Sqrt[1 + x^2 + x^4])

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Rubi in Sympy [A]  time = 37.5185, size = 172, normalized size = 0.94 \[ \frac{x^{3} \left (x^{4} + x^{2} + 1\right )^{\frac{3}{2}}}{9} + \frac{x \left (4 x^{2} + \frac{14}{3}\right ) \sqrt{x^{4} + x^{2} + 1}}{15} + \frac{x \left (x^{4} + x^{2} + 1\right )^{\frac{3}{2}}}{3} + \frac{26 x \sqrt{x^{4} + x^{2} + 1}}{45 \left (x^{2} + 1\right )} - \frac{26 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{45 \sqrt{x^{4} + x^{2} + 1}} + \frac{7 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{15 \sqrt{x^{4} + x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*(x**4 + x**2 + 1)**(3/2)/9 + x*(4*x**2 + 14/3)*sqrt(x**4 + x**2 + 1)/15 + x
*(x**4 + x**2 + 1)**(3/2)/3 + 26*x*sqrt(x**4 + x**2 + 1)/(45*(x**2 + 1)) - 26*sq
rt((x**4 + x**2 + 1)/(x**2 + 1)**2)*(x**2 + 1)*elliptic_e(2*atan(x), 1/4)/(45*sq
rt(x**4 + x**2 + 1)) + 7*sqrt((x**4 + x**2 + 1)/(x**2 + 1)**2)*(x**2 + 1)*ellipt
ic_f(2*atan(x), 1/4)/(15*sqrt(x**4 + x**2 + 1))

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Mathematica [C]  time = 0.372671, size = 169, normalized size = 0.92 \[ \frac{2 (-1)^{5/6} \left (4 \sqrt{3}+9 i\right ) \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+26 \sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+x \left (5 x^{10}+25 x^8+57 x^6+81 x^4+61 x^2+29\right )}{45 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(29 + 61*x^2 + 81*x^4 + 57*x^6 + 25*x^8 + 5*x^10) + 26*(-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)^(5/6)*(9*I + 4*Sqrt[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)])/(45*Sqrt[1 + x^2 + x^4]
)

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Maple [C]  time = 0.226, size = 263, normalized size = 1.4 \[{\frac{29\,x}{45}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{32}{45\,\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}-{\frac{104}{45\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}+{\frac{{x}^{7}}{9}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{4\,{x}^{5}}{9}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{32\,{x}^{3}}{45}\sqrt{{x}^{4}+{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

29/45*x*(x^4+x^2+1)^(1/2)+32/45/(-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))-104/45/(-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/
9*x^7*(x^4+x^2+1)^(1/2)+4/9*x^5*(x^4+x^2+1)^(1/2)+32/45*x^3*(x^4+x^2+1)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )}^{3}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((x^6 + 3*x^4 + 3*x^2 + 1)*sqrt(x^4 + x^2 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((x**2 - x + 1)*(x**2 + x + 1))*(x**2 + 1)**3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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