3.300 \(\int \left (2+3 x^2+x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=172 \[ \frac{1}{7} x \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{35} x \left (9 x^2+29\right ) \sqrt{x^4+3 x^2+2}+\frac{6 x \left (x^2+2\right )}{5 \sqrt{x^4+3 x^2+2}}+\frac{31 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{35 \sqrt{x^4+3 x^2+2}}-\frac{6 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{x^4+3 x^2+2}} \]

[Out]

(6*x*(2 + x^2))/(5*Sqrt[2 + 3*x^2 + x^4]) + (x*(29 + 9*x^2)*Sqrt[2 + 3*x^2 + x^4
])/35 + (x*(2 + 3*x^2 + x^4)^(3/2))/7 - (6*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 +
x^2)]*EllipticE[ArcTan[x], 1/2])/(5*Sqrt[2 + 3*x^2 + x^4]) + (31*Sqrt[2]*(1 + x
^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(35*Sqrt[2 + 3*x^2 + x^
4])

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Rubi [A]  time = 0.13493, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357 \[ \frac{1}{7} x \left (x^4+3 x^2+2\right )^{3/2}+\frac{1}{35} x \left (9 x^2+29\right ) \sqrt{x^4+3 x^2+2}+\frac{6 x \left (x^2+2\right )}{5 \sqrt{x^4+3 x^2+2}}+\frac{31 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} F\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{35 \sqrt{x^4+3 x^2+2}}-\frac{6 \sqrt{2} \left (x^2+1\right ) \sqrt{\frac{x^2+2}{x^2+1}} E\left (\tan ^{-1}(x)|\frac{1}{2}\right )}{5 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

[In]  Int[(2 + 3*x^2 + x^4)^(3/2),x]

[Out]

(6*x*(2 + x^2))/(5*Sqrt[2 + 3*x^2 + x^4]) + (x*(29 + 9*x^2)*Sqrt[2 + 3*x^2 + x^4
])/35 + (x*(2 + 3*x^2 + x^4)^(3/2))/7 - (6*Sqrt[2]*(1 + x^2)*Sqrt[(2 + x^2)/(1 +
x^2)]*EllipticE[ArcTan[x], 1/2])/(5*Sqrt[2 + 3*x^2 + x^4]) + (31*Sqrt[2]*(1 + x
^2)*Sqrt[(2 + x^2)/(1 + x^2)]*EllipticF[ArcTan[x], 1/2])/(35*Sqrt[2 + 3*x^2 + x^
4])

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Rubi in Sympy [A]  time = 21.1195, size = 156, normalized size = 0.91 \[ \frac{3 x \left (2 x^{2} + 4\right )}{5 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{x \left (9 x^{2} + 29\right ) \sqrt{x^{4} + 3 x^{2} + 2}}{35} + \frac{x \left (x^{4} + 3 x^{2} + 2\right )^{\frac{3}{2}}}{7} - \frac{3 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) E\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{10 \sqrt{x^{4} + 3 x^{2} + 2}} + \frac{31 \sqrt{\frac{2 x^{2} + 4}{x^{2} + 1}} \left (4 x^{2} + 4\right ) F\left (\operatorname{atan}{\left (x \right )}\middle | \frac{1}{2}\right )}{140 \sqrt{x^{4} + 3 x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*x*(2*x**2 + 4)/(5*sqrt(x**4 + 3*x**2 + 2)) + x*(9*x**2 + 29)*sqrt(x**4 + 3*x**
2 + 2)/35 + x*(x**4 + 3*x**2 + 2)**(3/2)/7 - 3*sqrt((2*x**2 + 4)/(x**2 + 1))*(4*
x**2 + 4)*elliptic_e(atan(x), 1/2)/(10*sqrt(x**4 + 3*x**2 + 2)) + 31*sqrt((2*x**
2 + 4)/(x**2 + 1))*(4*x**2 + 4)*elliptic_f(atan(x), 1/2)/(140*sqrt(x**4 + 3*x**2
+ 2))

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Mathematica [C]  time = 0.0665449, size = 114, normalized size = 0.66 \[ \frac{5 x^9+39 x^7+121 x^5+165 x^3-20 i \sqrt{x^2+1} \sqrt{x^2+2} F\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )-42 i \sqrt{x^2+1} \sqrt{x^2+2} E\left (\left .i \sinh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right |2\right )+78 x}{35 \sqrt{x^4+3 x^2+2}} \]

Antiderivative was successfully verified.

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

[Out]

(78*x + 165*x^3 + 121*x^5 + 39*x^7 + 5*x^9 - (42*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*
EllipticE[I*ArcSinh[x/Sqrt[2]], 2] - (20*I)*Sqrt[1 + x^2]*Sqrt[2 + x^2]*Elliptic
F[I*ArcSinh[x/Sqrt[2]], 2])/(35*Sqrt[2 + 3*x^2 + x^4])

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Maple [C]  time = 0.005, size = 155, normalized size = 0.9 \[{\frac{{x}^{5}}{7}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{24\,{x}^{3}}{35}\sqrt{{x}^{4}+3\,{x}^{2}+2}}+{\frac{39\,x}{35}\sqrt{{x}^{4}+3\,{x}^{2}+2}}-{{\frac{31\,i}{35}}\sqrt{2}{\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}}+{{\frac{3\,i}{5}}\sqrt{2} \left ({\it EllipticF} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) -{\it EllipticE} \left ({\frac{i}{2}}\sqrt{2}x,\sqrt{2} \right ) \right ) \sqrt{2\,{x}^{2}+4}\sqrt{{x}^{2}+1}{\frac{1}{\sqrt{{x}^{4}+3\,{x}^{2}+2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^4+3*x^2+2)^(3/2),x)

[Out]

1/7*x^5*(x^4+3*x^2+2)^(1/2)+24/35*x^3*(x^4+3*x^2+2)^(1/2)+39/35*x*(x^4+3*x^2+2)^
(1/2)-31/35*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^2+2)^(1/2)*Elliptic
F(1/2*I*2^(1/2)*x,2^(1/2))+3/5*I*2^(1/2)*(2*x^2+4)^(1/2)*(x^2+1)^(1/2)/(x^4+3*x^
2+2)^(1/2)*(EllipticF(1/2*I*2^(1/2)*x,2^(1/2))-EllipticE(1/2*I*2^(1/2)*x,2^(1/2)
))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^4 + 3*x^2 + 2)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((x^4 + 3*x^2 + 2)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x**4 + 3*x**2 + 2)**(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^4 + 3*x^2 + 2)^(3/2), x)