3.141 \(\int \frac{11 x+2 x^3}{\left (3+2 x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=45 \[ \frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{9 x^2+5}{8 \left (x^4+2 x^2+3\right )} \]

[Out]

(5 + 9*x^2)/(8*(3 + 2*x^2 + x^4)) + (9*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2])

_______________________________________________________________________________________

Rubi [A]  time = 0.0907152, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ \frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{9 x^2+5}{8 \left (x^4+2 x^2+3\right )} \]

Antiderivative was successfully verified.

[In]  Int[(11*x + 2*x^3)/(3 + 2*x^2 + x^4)^2,x]

[Out]

(5 + 9*x^2)/(8*(3 + 2*x^2 + x^4)) + (9*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.9234, size = 41, normalized size = 0.91 \[ \frac{18 x^{2} + 10}{16 \left (x^{4} + 2 x^{2} + 3\right )} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{x^{2}}{2} + \frac{1}{2}\right ) \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2*x**3+11*x)/(x**4+2*x**2+3)**2,x)

[Out]

(18*x**2 + 10)/(16*(x**4 + 2*x**2 + 3)) + 9*sqrt(2)*atan(sqrt(2)*(x**2/2 + 1/2))
/16

_______________________________________________________________________________________

Mathematica [A]  time = 0.0465281, size = 45, normalized size = 1. \[ \frac{9 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}}+\frac{9 x^2+5}{8 \left (x^4+2 x^2+3\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(11*x + 2*x^3)/(3 + 2*x^2 + x^4)^2,x]

[Out]

(5 + 9*x^2)/(8*(3 + 2*x^2 + x^4)) + (9*ArcTan[(1 + x^2)/Sqrt[2]])/(8*Sqrt[2])

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 41, normalized size = 0.9 \[{\frac{18\,{x}^{2}+10}{16\,{x}^{4}+32\,{x}^{2}+48}}+{\frac{9\,\sqrt{2}}{16}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2*x^3+11*x)/(x^4+2*x^2+3)^2,x)

[Out]

1/16*(18*x^2+10)/(x^4+2*x^2+3)+9/16*2^(1/2)*arctan(1/4*(2*x^2+2)*2^(1/2))

_______________________________________________________________________________________

Maxima [A]  time = 0.82534, size = 51, normalized size = 1.13 \[ \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{9 \, x^{2} + 5}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^3 + 11*x)/(x^4 + 2*x^2 + 3)^2,x, algorithm="maxima")

[Out]

9/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 1/8*(9*x^2 + 5)/(x^4 + 2*x^2 + 3)

_______________________________________________________________________________________

Fricas [A]  time = 0.281232, size = 70, normalized size = 1.56 \[ \frac{\sqrt{2}{\left (9 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \sqrt{2}{\left (9 \, x^{2} + 5\right )}\right )}}{16 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^3 + 11*x)/(x^4 + 2*x^2 + 3)^2,x, algorithm="fricas")

[Out]

1/16*sqrt(2)*(9*(x^4 + 2*x^2 + 3)*arctan(1/2*sqrt(2)*(x^2 + 1)) + sqrt(2)*(9*x^2
+ 5))/(x^4 + 2*x^2 + 3)

_______________________________________________________________________________________

Sympy [A]  time = 0.376625, size = 44, normalized size = 0.98 \[ \frac{9 x^{2} + 5}{8 x^{4} + 16 x^{2} + 24} + \frac{9 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x**3+11*x)/(x**4+2*x**2+3)**2,x)

[Out]

(9*x**2 + 5)/(8*x**4 + 16*x**2 + 24) + 9*sqrt(2)*atan(sqrt(2)*x**2/2 + sqrt(2)/2
)/16

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.27076, size = 51, normalized size = 1.13 \[ \frac{9}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + \frac{9 \, x^{2} + 5}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x^3 + 11*x)/(x^4 + 2*x^2 + 3)^2,x, algorithm="giac")

[Out]

9/16*sqrt(2)*arctan(1/2*sqrt(2)*(x^2 + 1)) + 1/8*(9*x^2 + 5)/(x^4 + 2*x^2 + 3)