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

Optimal. Leaf size=72 \[ \frac{x \left (25 x^2+24\right )}{4 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (130 x^2+211\right )}{8 \left (x^4+3 x^2+2\right )}+\frac{317}{8} \tan ^{-1}(x)-\frac{447 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{8 \sqrt{2}} \]

[Out]

(x*(24 + 25*x^2))/(4*(2 + 3*x^2 + x^4)^2) - (x*(211 + 130*x^2))/(8*(2 + 3*x^2 +
x^4)) + (317*ArcTan[x])/8 - (447*ArcTan[x/Sqrt[2]])/(8*Sqrt[2])

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Rubi [A]  time = 0.118906, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129 \[ \frac{x \left (25 x^2+24\right )}{4 \left (x^4+3 x^2+2\right )^2}-\frac{x \left (130 x^2+211\right )}{8 \left (x^4+3 x^2+2\right )}+\frac{317}{8} \tan ^{-1}(x)-\frac{447 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{8 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(24 + 25*x^2))/(4*(2 + 3*x^2 + x^4)^2) - (x*(211 + 130*x^2))/(8*(2 + 3*x^2 +
x^4)) + (317*ArcTan[x])/8 - (447*ArcTan[x/Sqrt[2]])/(8*Sqrt[2])

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Rubi in Sympy [A]  time = 19.4126, size = 66, normalized size = 0.92 \[ \frac{x \left (4050 x^{2} + 3888\right )}{648 \left (x^{4} + 3 x^{2} + 2\right )^{2}} - \frac{x \left (379080 x^{2} + 615276\right )}{23328 \left (x^{4} + 3 x^{2} + 2\right )} + \frac{317 \operatorname{atan}{\left (x \right )}}{8} - \frac{447 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{16} \]

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)**3,x)

[Out]

x*(4050*x**2 + 3888)/(648*(x**4 + 3*x**2 + 2)**2) - x*(379080*x**2 + 615276)/(23
328*(x**4 + 3*x**2 + 2)) + 317*atan(x)/8 - 447*sqrt(2)*atan(sqrt(2)*x/2)/16

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Mathematica [A]  time = 0.104772, size = 56, normalized size = 0.78 \[ \frac{1}{16} \left (-\frac{2 x \left (130 x^6+601 x^4+843 x^2+374\right )}{\left (x^4+3 x^2+2\right )^2}+634 \tan ^{-1}(x)-447 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-2*x*(374 + 843*x^2 + 601*x^4 + 130*x^6))/(2 + 3*x^2 + x^4)^2 + 634*ArcTan[x]
- 447*Sqrt[2]*ArcTan[x/Sqrt[2]])/16

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Maple [A]  time = 0.017, size = 53, normalized size = 0.7 \[ -{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ({\frac{103\,{x}^{3}}{8}}+{\frac{129\,x}{4}} \right ) }-{\frac{447\,\sqrt{2}}{16}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{27\,{x}^{3}}{8}}-{\frac{29\,x}{8}} \right ) }+{\frac{317\,\arctan \left ( x \right ) }{8}} \]

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)^3,x)

[Out]

-(103/8*x^3+129/4*x)/(x^2+2)^2-447/16*arctan(1/2*2^(1/2)*x)*2^(1/2)+(-27/8*x^3-2
9/8*x)/(x^2+1)^2+317/8*arctan(x)

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Maxima [A]  time = 0.791054, size = 81, normalized size = 1.12 \[ -\frac{447}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{130 \, x^{7} + 601 \, x^{5} + 843 \, x^{3} + 374 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} + \frac{317}{8} \, \arctan \left (x\right ) \]

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)^3,x, algorithm="maxima")

[Out]

-447/16*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/8*(130*x^7 + 601*x^5 + 843*x^3 + 374*x
)/(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4) + 317/8*arctan(x)

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Fricas [A]  time = 0.272438, size = 146, normalized size = 2.03 \[ \frac{\sqrt{2}{\left (317 \, \sqrt{2}{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) - 447 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \sqrt{2}{\left (130 \, x^{7} + 601 \, x^{5} + 843 \, x^{3} + 374 \, x\right )}\right )}}{16 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

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)^3,x, algorithm="fricas")

[Out]

1/16*sqrt(2)*(317*sqrt(2)*(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*arctan(x) - 447*(x
^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*arctan(1/2*sqrt(2)*x) - sqrt(2)*(130*x^7 + 601
*x^5 + 843*x^3 + 374*x))/(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)

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Sympy [A]  time = 0.795633, size = 65, normalized size = 0.9 \[ - \frac{130 x^{7} + 601 x^{5} + 843 x^{3} + 374 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} + \frac{317 \operatorname{atan}{\left (x \right )}}{8} - \frac{447 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{16} \]

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)**3,x)

[Out]

-(130*x**7 + 601*x**5 + 843*x**3 + 374*x)/(8*x**8 + 48*x**6 + 104*x**4 + 96*x**2
+ 32) + 317*atan(x)/8 - 447*sqrt(2)*atan(sqrt(2)*x/2)/16

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GIAC/XCAS [A]  time = 0.271162, size = 68, normalized size = 0.94 \[ -\frac{447}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{130 \, x^{7} + 601 \, x^{5} + 843 \, x^{3} + 374 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac{317}{8} \, \arctan \left (x\right ) \]

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)^3,x, algorithm="giac")

[Out]

-447/16*sqrt(2)*arctan(1/2*sqrt(2)*x) - 1/8*(130*x^7 + 601*x^5 + 843*x^3 + 374*x
)/(x^4 + 3*x^2 + 2)^2 + 317/8*arctan(x)