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

Optimal. Leaf size=159 \[ \frac{14 \sqrt{x^4+x^2+1} x}{15 \left (x^2+1\right )}+\frac{11}{15} \sqrt{x^4+x^2+1} x+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{5 \sqrt{x^4+x^2+1}}-\frac{14 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{15 \sqrt{x^4+x^2+1}}+\frac{1}{5} \sqrt{x^4+x^2+1} x^3 \]

[Out]

(11*x*Sqrt[1 + x^2 + x^4])/15 + (x^3*Sqrt[1 + x^2 + x^4])/5 + (14*x*Sqrt[1 + x^2
+ x^4])/(15*(1 + x^2)) - (14*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*Ellipt
icE[2*ArcTan[x], 1/4])/(15*Sqrt[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 = 0.14811, antiderivative size = 159, 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 \[ \frac{14 \sqrt{x^4+x^2+1} x}{15 \left (x^2+1\right )}+\frac{11}{15} \sqrt{x^4+x^2+1} x+\frac{3 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{5 \sqrt{x^4+x^2+1}}-\frac{14 \left (x^2+1\right ) \sqrt{\frac{x^4+x^2+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac{1}{4}\right )}{15 \sqrt{x^4+x^2+1}}+\frac{1}{5} \sqrt{x^4+x^2+1} x^3 \]

Antiderivative was successfully verified.

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

[Out]

(11*x*Sqrt[1 + x^2 + x^4])/15 + (x^3*Sqrt[1 + x^2 + x^4])/5 + (14*x*Sqrt[1 + x^2
+ x^4])/(15*(1 + x^2)) - (14*(1 + x^2)*Sqrt[(1 + x^2 + x^4)/(1 + x^2)^2]*Ellipt
icE[2*ArcTan[x], 1/4])/(15*Sqrt[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 = 30.5789, size = 150, normalized size = 0.94 \[ \frac{x^{3} \sqrt{x^{4} + x^{2} + 1}}{5} + \frac{11 x \sqrt{x^{4} + x^{2} + 1}}{15} + \frac{14 x \sqrt{x^{4} + x^{2} + 1}}{15 \left (x^{2} + 1\right )} - \frac{14 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) E\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{15 \sqrt{x^{4} + x^{2} + 1}} + \frac{3 \sqrt{\frac{x^{4} + x^{2} + 1}{\left (x^{2} + 1\right )^{2}}} \left (x^{2} + 1\right ) F\left (2 \operatorname{atan}{\left (x \right )}\middle | \frac{1}{4}\right )}{5 \sqrt{x^{4} + x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**3*sqrt(x**4 + x**2 + 1)/5 + 11*x*sqrt(x**4 + x**2 + 1)/15 + 14*x*sqrt(x**4 +
x**2 + 1)/(15*(x**2 + 1)) - 14*sqrt((x**4 + x**2 + 1)/(x**2 + 1)**2)*(x**2 + 1)*
elliptic_e(2*atan(x), 1/4)/(15*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 = 0.219053, size = 157, normalized size = 0.99 \[ \frac{2 \sqrt [3]{-1} \left (2 \sqrt [3]{-1}-7\right ) \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} F\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+14 \sqrt [3]{-1} \sqrt{\sqrt [3]{-1} x^2+1} \sqrt{1-(-1)^{2/3} x^2} E\left (i \sinh ^{-1}\left ((-1)^{5/6} x\right )|(-1)^{2/3}\right )+x \left (3 x^6+14 x^4+14 x^2+11\right )}{15 \sqrt{x^4+x^2+1}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(11 + 14*x^2 + 14*x^4 + 3*x^6) + 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 + 2*(-1)^(1/3))*Sqrt[1 + (-1)^(1/3)*x^2]*Sqrt[1 - (-1)^(2/3)*x^2]*Elliptic
F[I*ArcSinh[(-1)^(5/6)*x], (-1)^(2/3)])/(15*Sqrt[1 + x^2 + x^4])

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Maple [C]  time = 0.037, size = 233, normalized size = 1.5 \[{\frac{8}{15\,\sqrt{-2+2\,i\sqrt{3}}}\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}{\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}}+{\frac{{x}^{3}}{5}\sqrt{{x}^{4}+{x}^{2}+1}}+{\frac{11\,x}{15}\sqrt{{x}^{4}+{x}^{2}+1}}-{\frac{56}{15\,\sqrt{-2+2\,i\sqrt{3}} \left ( i\sqrt{3}+1 \right ) }\sqrt{1- \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){x}^{2}}\sqrt{1- \left ( -{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ){x}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) -{\it EllipticE} \left ({\frac{x\sqrt{-2+2\,i\sqrt{3}}}{2}},{\frac{\sqrt{-2+2\,i\sqrt{3}}}{2}} \right ) \right ){\frac{1}{\sqrt{{x}^{4}+{x}^{2}+1}}}} \]

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]

8/15/(-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))+1/5*x^3*(x^4+x^2+1)^(1/2)+11/15*x*(x^4+x^2+1)^(1/2)-56/15
/(-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)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 + 1)^3/sqrt(x^4 + x^2 + 1),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. \[ \int \frac{\left (x^{2} + 1\right )^{3}}{\sqrt{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right )}}\, dx \]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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