3.92 \(\int \frac{x^8 \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx\)

Optimal. Leaf size=80 \[ \frac{5 x^3}{3}+\frac{\left (24-409 x^2\right ) x}{8 \left (x^4+3 x^2+2\right )}-\frac{\left (207 x^2+206\right ) x}{4 \left (x^4+3 x^2+2\right )^2}-42 x-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

[Out]

-42*x + (5*x^3)/3 - (x*(206 + 207*x^2))/(4*(2 + 3*x^2 + x^4)^2) + (x*(24 - 409*x
^2))/(8*(2 + 3*x^2 + x^4)) - (449*ArcTan[x])/8 + (219*ArcTan[x/Sqrt[2]])/Sqrt[2]

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Rubi [A]  time = 0.165412, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161 \[ \frac{5 x^3}{3}+\frac{\left (24-409 x^2\right ) x}{8 \left (x^4+3 x^2+2\right )}-\frac{\left (207 x^2+206\right ) x}{4 \left (x^4+3 x^2+2\right )^2}-42 x-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Int[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]

[Out]

-42*x + (5*x^3)/3 - (x*(206 + 207*x^2))/(4*(2 + 3*x^2 + x^4)^2) + (x*(24 - 409*x
^2))/(8*(2 + 3*x^2 + x^4)) - (449*ArcTan[x])/8 + (219*ArcTan[x/Sqrt[2]])/Sqrt[2]

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Rubi in Sympy [A]  time = 32.119, size = 76, normalized size = 0.95 \[ \frac{5 x^{3}}{3} + \frac{x \left (- 869437476 x^{2} + 51018336\right )}{17006112 \left (x^{4} + 3 x^{2} + 2\right )} - \frac{x \left (905418 x^{2} + 901044\right )}{17496 \left (x^{4} + 3 x^{2} + 2\right )^{2}} - 42 x - \frac{449 \operatorname{atan}{\left (x \right )}}{8} + \frac{219 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**3,x)

[Out]

5*x**3/3 + x*(-869437476*x**2 + 51018336)/(17006112*(x**4 + 3*x**2 + 2)) - x*(90
5418*x**2 + 901044)/(17496*(x**4 + 3*x**2 + 2)**2) - 42*x - 449*atan(x)/8 + 219*
sqrt(2)*atan(sqrt(2)*x/2)/2

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Mathematica [A]  time = 0.100847, size = 66, normalized size = 0.82 \[ \frac{x \left (40 x^{10}-768 x^8-6755 x^6-16233 x^4-15416 x^2-5124\right )}{24 \left (x^4+3 x^2+2\right )^2}-\frac{449}{8} \tan ^{-1}(x)+\frac{219 \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]

[Out]

(x*(-5124 - 15416*x^2 - 16233*x^4 - 6755*x^6 - 768*x^8 + 40*x^10))/(24*(2 + 3*x^
2 + x^4)^2) - (449*ArcTan[x])/8 + (219*ArcTan[x/Sqrt[2]])/Sqrt[2]

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Maple [A]  time = 0.018, size = 62, normalized size = 0.8 \[{\frac{5\,{x}^{3}}{3}}-42\,x+16\,{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ( -{\frac{53\,{x}^{3}}{16}}-{\frac{27\,x}{8}} \right ) }+{\frac{219\,\sqrt{2}}{2}\arctan \left ({\frac{\sqrt{2}x}{2}} \right ) }-{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{15\,{x}^{3}}{8}}-{\frac{17\,x}{8}} \right ) }-{\frac{449\,\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^3,x)

[Out]

5/3*x^3-42*x+16*(-53/16*x^3-27/8*x)/(x^2+2)^2+219/2*arctan(1/2*2^(1/2)*x)*2^(1/2
)-(-15/8*x^3-17/8*x)/(x^2+1)^2-449/8*arctan(x)

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Maxima [A]  time = 0.796956, size = 92, normalized size = 1.15 \[ \frac{5}{3} \, x^{3} + \frac{219}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 42 \, x - \frac{409 \, x^{7} + 1203 \, x^{5} + 1160 \, x^{3} + 364 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} - \frac{449}{8} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 3*x^2 + 2)^3,x, algorithm="maxima")

[Out]

5/3*x^3 + 219/2*sqrt(2)*arctan(1/2*sqrt(2)*x) - 42*x - 1/8*(409*x^7 + 1203*x^5 +
1160*x^3 + 364*x)/(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4) - 449/8*arctan(x)

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Fricas [A]  time = 0.284211, size = 159, normalized size = 1.99 \[ -\frac{\sqrt{2}{\left (1347 \, \sqrt{2}{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) - 5256 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \sqrt{2}{\left (40 \, x^{11} - 768 \, x^{9} - 6755 \, x^{7} - 16233 \, x^{5} - 15416 \, x^{3} - 5124 \, x\right )}\right )}}{48 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 3*x^2 + 2)^3,x, algorithm="fricas")

[Out]

-1/48*sqrt(2)*(1347*sqrt(2)*(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*arctan(x) - 5256
*(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*arctan(1/2*sqrt(2)*x) - sqrt(2)*(40*x^11 -
768*x^9 - 6755*x^7 - 16233*x^5 - 15416*x^3 - 5124*x))/(x^8 + 6*x^6 + 13*x^4 + 12
*x^2 + 4)

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Sympy [A]  time = 0.774693, size = 75, normalized size = 0.94 \[ \frac{5 x^{3}}{3} - 42 x - \frac{409 x^{7} + 1203 x^{5} + 1160 x^{3} + 364 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} - \frac{449 \operatorname{atan}{\left (x \right )}}{8} + \frac{219 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**3,x)

[Out]

5*x**3/3 - 42*x - (409*x**7 + 1203*x**5 + 1160*x**3 + 364*x)/(8*x**8 + 48*x**6 +
104*x**4 + 96*x**2 + 32) - 449*atan(x)/8 + 219*sqrt(2)*atan(sqrt(2)*x/2)/2

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GIAC/XCAS [A]  time = 0.271521, size = 78, normalized size = 0.98 \[ \frac{5}{3} \, x^{3} + \frac{219}{2} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - 42 \, x - \frac{409 \, x^{7} + 1203 \, x^{5} + 1160 \, x^{3} + 364 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} - \frac{449}{8} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 3*x^2 + 2)^3,x, algorithm="giac")

[Out]

5/3*x^3 + 219/2*sqrt(2)*arctan(1/2*sqrt(2)*x) - 42*x - 1/8*(409*x^7 + 1203*x^5 +
1160*x^3 + 364*x)/(x^4 + 3*x^2 + 2)^2 - 449/8*arctan(x)