3.91 \(\int \frac{x^{10} \left (4+x^2+3 x^4+5 x^6\right )}{\left (2+3 x^2+x^4\right )^3} \, dx\)

Optimal. Leaf size=81 \[ x^5-14 x^3+\frac{\left (1669 x^2+824\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac{\left (415 x^2+414\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+214 x+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

[Out]

214*x - 14*x^3 + x^5 + (x*(414 + 415*x^2))/(4*(2 + 3*x^2 + x^4)^2) + (x*(824 + 1
669*x^2))/(8*(2 + 3*x^2 + x^4)) + (477*ArcTan[x])/8 - 351*Sqrt[2]*ArcTan[x/Sqrt[
2]]

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Rubi [A]  time = 0.177441, antiderivative size = 81, 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 \[ x^5-14 x^3+\frac{\left (1669 x^2+824\right ) x}{8 \left (x^4+3 x^2+2\right )}+\frac{\left (415 x^2+414\right ) x}{4 \left (x^4+3 x^2+2\right )^2}+214 x+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x^10*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]

[Out]

214*x - 14*x^3 + x^5 + (x*(414 + 415*x^2))/(4*(2 + 3*x^2 + x^4)^2) + (x*(824 + 1
669*x^2))/(8*(2 + 3*x^2 + x^4)) + (477*ArcTan[x])/8 - 351*Sqrt[2]*ArcTan[x/Sqrt[
2]]

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Rubi in Sympy [A]  time = 37.3971, size = 76, normalized size = 0.94 \[ x^{5} - 14 x^{3} + \frac{x \left (5445630 x^{2} + 5432508\right )}{52488 \left (x^{4} + 3 x^{2} + 2\right )^{2}} + \frac{x \left (31931101044 x^{2} + 15764665824\right )}{153055008 \left (x^{4} + 3 x^{2} + 2\right )} + 214 x + \frac{477 \operatorname{atan}{\left (x \right )}}{8} - 351 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**10*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**3,x)

[Out]

x**5 - 14*x**3 + x*(5445630*x**2 + 5432508)/(52488*(x**4 + 3*x**2 + 2)**2) + x*(
31931101044*x**2 + 15764665824)/(153055008*(x**4 + 3*x**2 + 2)) + 214*x + 477*at
an(x)/8 - 351*sqrt(2)*atan(sqrt(2)*x/2)

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Mathematica [A]  time = 0.120437, size = 71, normalized size = 0.88 \[ \frac{x \left (8 x^{12}-64 x^{10}+1144 x^8+10581 x^6+26775 x^4+26736 x^2+9324\right )}{8 \left (x^4+3 x^2+2\right )^2}+\frac{477}{8} \tan ^{-1}(x)-351 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x^10*(4 + x^2 + 3*x^4 + 5*x^6))/(2 + 3*x^2 + x^4)^3,x]

[Out]

(x*(9324 + 26736*x^2 + 26775*x^4 + 10581*x^6 + 1144*x^8 - 64*x^10 + 8*x^12))/(8*
(2 + 3*x^2 + x^4)^2) + (477*ArcTan[x])/8 - 351*Sqrt[2]*ArcTan[x/Sqrt[2]]

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Maple [A]  time = 0.018, size = 64, normalized size = 0.8 \[{x}^{5}-14\,{x}^{3}+214\,x-16\,{\frac{1}{ \left ({x}^{2}+2 \right ) ^{2}} \left ( -{\frac{105\,{x}^{3}}{8}}-{\frac{79\,x}{4}} \right ) }-351\,\arctan \left ( 1/2\,\sqrt{2}x \right ) \sqrt{2}+{\frac{1}{ \left ({x}^{2}+1 \right ) ^{2}} \left ( -{\frac{11\,{x}^{3}}{8}}-{\frac{13\,x}{8}} \right ) }+{\frac{477\,\arctan \left ( x \right ) }{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^10*(5*x^6+3*x^4+x^2+4)/(x^4+3*x^2+2)^3,x)

[Out]

x^5-14*x^3+214*x-16*(-105/8*x^3-79/4*x)/(x^2+2)^2-351*arctan(1/2*2^(1/2)*x)*2^(1
/2)+(-11/8*x^3-13/8*x)/(x^2+1)^2+477/8*arctan(x)

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Maxima [A]  time = 0.796205, size = 96, normalized size = 1.19 \[ x^{5} - 14 \, x^{3} - 351 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 214 \, x + \frac{1669 \, x^{7} + 5831 \, x^{5} + 6640 \, x^{3} + 2476 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} + \frac{477}{8} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^10/(x^4 + 3*x^2 + 2)^3,x, algorithm="maxima")

[Out]

x^5 - 14*x^3 - 351*sqrt(2)*arctan(1/2*sqrt(2)*x) + 214*x + 1/8*(1669*x^7 + 5831*
x^5 + 6640*x^3 + 2476*x)/(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4) + 477/8*arctan(x)

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Fricas [A]  time = 0.264596, size = 154, normalized size = 1.9 \[ \frac{8 \, x^{13} - 64 \, x^{11} + 1144 \, x^{9} + 10581 \, x^{7} + 26775 \, x^{5} + 26736 \, x^{3} - 2808 \, \sqrt{2}{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 477 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )} \arctan \left (x\right ) + 9324 \, x}{8 \,{\left (x^{8} + 6 \, x^{6} + 13 \, x^{4} + 12 \, x^{2} + 4\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^10/(x^4 + 3*x^2 + 2)^3,x, algorithm="fricas")

[Out]

1/8*(8*x^13 - 64*x^11 + 1144*x^9 + 10581*x^7 + 26775*x^5 + 26736*x^3 - 2808*sqrt
(2)*(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)*arctan(1/2*sqrt(2)*x) + 477*(x^8 + 6*x^6
+ 13*x^4 + 12*x^2 + 4)*arctan(x) + 9324*x)/(x^8 + 6*x^6 + 13*x^4 + 12*x^2 + 4)

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Sympy [A]  time = 0.800088, size = 75, normalized size = 0.93 \[ x^{5} - 14 x^{3} + 214 x + \frac{1669 x^{7} + 5831 x^{5} + 6640 x^{3} + 2476 x}{8 x^{8} + 48 x^{6} + 104 x^{4} + 96 x^{2} + 32} + \frac{477 \operatorname{atan}{\left (x \right )}}{8} - 351 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**10*(5*x**6+3*x**4+x**2+4)/(x**4+3*x**2+2)**3,x)

[Out]

x**5 - 14*x**3 + 214*x + (1669*x**7 + 5831*x**5 + 6640*x**3 + 2476*x)/(8*x**8 +
48*x**6 + 104*x**4 + 96*x**2 + 32) + 477*atan(x)/8 - 351*sqrt(2)*atan(sqrt(2)*x/
2)

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GIAC/XCAS [A]  time = 0.271586, size = 82, normalized size = 1.01 \[ x^{5} - 14 \, x^{3} - 351 \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) + 214 \, x + \frac{1669 \, x^{7} + 5831 \, x^{5} + 6640 \, x^{3} + 2476 \, x}{8 \,{\left (x^{4} + 3 \, x^{2} + 2\right )}^{2}} + \frac{477}{8} \, \arctan \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^10/(x^4 + 3*x^2 + 2)^3,x, algorithm="giac")

[Out]

x^5 - 14*x^3 - 351*sqrt(2)*arctan(1/2*sqrt(2)*x) + 214*x + 1/8*(1669*x^7 + 5831*
x^5 + 6640*x^3 + 2476*x)/(x^4 + 3*x^2 + 2)^2 + 477/8*arctan(x)