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

Optimal. Leaf size=44 \[ \frac{1}{10} \left (x^2+1\right )^2 \left (x^4+2 x^2+2\right )^{3/2}-\frac{1}{15} \left (x^4+2 x^2+2\right )^{3/2} \]

[Out]

-(2 + 2*x^2 + x^4)^(3/2)/15 + ((1 + x^2)^2*(2 + 2*x^2 + x^4)^(3/2))/10

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Rubi [A]  time = 0.0948337, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{1}{10} \left (x^2+1\right )^2 \left (x^4+2 x^2+2\right )^{3/2}-\frac{1}{15} \left (x^4+2 x^2+2\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

-(2 + 2*x^2 + x^4)^(3/2)/15 + ((1 + x^2)^2*(2 + 2*x^2 + x^4)^(3/2))/10

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Rubi in Sympy [A]  time = 6.28886, size = 36, normalized size = 0.82 \[ \frac{\left (x^{2} + 1\right )^{2} \left (x^{4} + 2 x^{2} + 2\right )^{\frac{3}{2}}}{10} - \frac{\left (x^{4} + 2 x^{2} + 2\right )^{\frac{3}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x**2 + 1)**2*(x**4 + 2*x**2 + 2)**(3/2)/10 - (x**4 + 2*x**2 + 2)**(3/2)/15

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Mathematica [A]  time = 0.0211666, size = 30, normalized size = 0.68 \[ \frac{1}{30} \left (x^4+2 x^2+2\right )^{3/2} \left (3 x^4+6 x^2+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

((2 + 2*x^2 + x^4)^(3/2)*(1 + 6*x^2 + 3*x^4))/30

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Maple [A]  time = 0.009, size = 27, normalized size = 0.6 \[{\frac{3\,{x}^{4}+6\,{x}^{2}+1}{30} \left ({x}^{4}+2\,{x}^{2}+2 \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(x^4+2*x^2+2)^(3/2)*(3*x^4+6*x^2+1)

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Maxima [A]  time = 0.806657, size = 66, normalized size = 1.5 \[ \frac{1}{10} \,{\left (x^{4} + 2 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{4} + \frac{1}{5} \,{\left (x^{4} + 2 \, x^{2} + 2\right )}^{\frac{3}{2}} x^{2} + \frac{1}{30} \,{\left (x^{4} + 2 \, x^{2} + 2\right )}^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*(x^4 + 2*x^2 + 2)^(3/2)*x^4 + 1/5*(x^4 + 2*x^2 + 2)^(3/2)*x^2 + 1/30*(x^4 +
2*x^2 + 2)^(3/2)

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Fricas [A]  time = 0.26269, size = 49, normalized size = 1.11 \[ \frac{1}{30} \,{\left (3 \, x^{8} + 12 \, x^{6} + 19 \, x^{4} + 14 \, x^{2} + 2\right )} \sqrt{x^{4} + 2 \, x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(3*x^8 + 12*x^6 + 19*x^4 + 14*x^2 + 2)*sqrt(x^4 + 2*x^2 + 2)

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Sympy [A]  time = 2.06952, size = 94, normalized size = 2.14 \[ \frac{x^{8} \sqrt{x^{4} + 2 x^{2} + 2}}{10} + \frac{2 x^{6} \sqrt{x^{4} + 2 x^{2} + 2}}{5} + \frac{19 x^{4} \sqrt{x^{4} + 2 x^{2} + 2}}{30} + \frac{7 x^{2} \sqrt{x^{4} + 2 x^{2} + 2}}{15} + \frac{\sqrt{x^{4} + 2 x^{2} + 2}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**8*sqrt(x**4 + 2*x**2 + 2)/10 + 2*x**6*sqrt(x**4 + 2*x**2 + 2)/5 + 19*x**4*sqr
t(x**4 + 2*x**2 + 2)/30 + 7*x**2*sqrt(x**4 + 2*x**2 + 2)/15 + sqrt(x**4 + 2*x**2
+ 2)/15

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GIAC/XCAS [A]  time = 0.262465, size = 51, normalized size = 1.16 \[ \frac{1}{30} \, \sqrt{x^{4} + 2 \, x^{2} + 2}{\left ({\left ({\left (3 \,{\left (x^{2} + 4\right )} x^{2} + 19\right )} x^{2} + 14\right )} x^{2} + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*sqrt(x^4 + 2*x^2 + 2)*(((3*(x^2 + 4)*x^2 + 19)*x^2 + 14)*x^2 + 2)