3.85 \(\int \frac{x^2 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{\left (25 x^2+24\right ) x}{2 \left (x^4+3 x^2+2\right )}+5 x-\frac{15}{2} \tan ^{-1}(x)-\frac{7 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

5*x + (x*(24 + 25*x^2))/(2*(2 + 3*x^2 + x^4)) - (15*ArcTan[x])/2 - (7*ArcTan[x/S
qrt[2]])/Sqrt[2]

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Rubi [A]  time = 0.114363, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{\left (25 x^2+24\right ) x}{2 \left (x^4+3 x^2+2\right )}+5 x-\frac{15}{2} \tan ^{-1}(x)-\frac{7 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^2*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]

[Out]

5*x + (x*(24 + 25*x^2))/(2*(2 + 3*x^2 + x^4)) - (15*ArcTan[x])/2 - (7*ArcTan[x/S
qrt[2]])/Sqrt[2]

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Rubi in Sympy [A]  time = 19.5035, size = 48, normalized size = 0.98 \[ \frac{x \left (4050 x^{2} + 3888\right )}{324 \left (x^{4} + 3 x^{2} + 2\right )} + 5 x - \frac{15 \operatorname{atan}{\left (x \right )}}{2} - \frac{7 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**2,x)

[Out]

x*(4050*x**2 + 3888)/(324*(x**4 + 3*x**2 + 2)) + 5*x - 15*atan(x)/2 - 7*sqrt(2)*
atan(sqrt(2)*x/2)/2

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Mathematica [A]  time = 0.0698881, size = 50, normalized size = 1.02 \[ \frac{25 x^3+24 x}{2 \left (x^4+3 x^2+2\right )}+5 x-\frac{15}{2} \tan ^{-1}(x)-\frac{7 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^2,x]

[Out]

5*x + (24*x + 25*x^3)/(2*(2 + 3*x^2 + x^4)) - (15*ArcTan[x])/2 - (7*ArcTan[x/Sqr
t[2]])/Sqrt[2]

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Maple [A]  time = 0.016, size = 41, normalized size = 0.8 \[ 5\,x+13\,{\frac{x}{{x}^{2}+2}}-{\frac{7\,\sqrt{2}}{2}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }-{\frac{x}{2\,{x}^{2}+2}}-{\frac{15\,\arctan \left ( x \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^2,x)

[Out]

5*x+13*x/(x^2+2)-7/2*arctan(1/2*2^(1/2)*x)*2^(1/2)-1/2*x/(x^2+1)-15/2*arctan(x)

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Maxima [A]  time = 0.793351, size = 58, normalized size = 1.18 \[ -\frac{7}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 5 \, x + \frac{25 \, x^{3} + 24 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac{15}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^2/(x^4 + 3*x^2 + 2)^2,x, algorithm="maxima")

[Out]

-7/2*sqrt(2)*arctan(1/2*sqrt(2)*x) + 5*x + 1/2*(25*x^3 + 24*x)/(x^4 + 3*x^2 + 2)
- 15/2*arctan(x)

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Fricas [A]  time = 0.271361, size = 99, normalized size = 2.02 \[ -\frac{\sqrt{2}{\left (15 \, \sqrt{2}{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (x\right ) + 14 \,{\left (x^{4} + 3 \, x^{2} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \sqrt{2}{\left (10 \, x^{5} + 55 \, x^{3} + 44 \, x\right )}\right )}}{4 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^2/(x^4 + 3*x^2 + 2)^2,x, algorithm="fricas")

[Out]

-1/4*sqrt(2)*(15*sqrt(2)*(x^4 + 3*x^2 + 2)*arctan(x) + 14*(x^4 + 3*x^2 + 2)*arct
an(1/2*sqrt(2)*x) - sqrt(2)*(10*x^5 + 55*x^3 + 44*x))/(x^4 + 3*x^2 + 2)

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Sympy [A]  time = 0.581369, size = 48, normalized size = 0.98 \[ 5 x + \frac{25 x^{3} + 24 x}{2 x^{4} + 6 x^{2} + 4} - \frac{15 \operatorname{atan}{\left (x \right )}}{2} - \frac{7 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**2,x)

[Out]

5*x + (25*x**3 + 24*x)/(2*x**4 + 6*x**2 + 4) - 15*atan(x)/2 - 7*sqrt(2)*atan(sqr
t(2)*x/2)/2

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GIAC/XCAS [A]  time = 0.276143, size = 58, normalized size = 1.18 \[ -\frac{7}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 5 \, x + \frac{25 \, x^{3} + 24 \, x}{2 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}} - \frac{15}{2} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^2/(x^4 + 3*x^2 + 2)^2,x, algorithm="giac")

[Out]

-7/2*sqrt(2)*arctan(1/2*sqrt(2)*x) + 5*x + 1/2*(25*x^3 + 24*x)/(x^4 + 3*x^2 + 2)
- 15/2*arctan(x)