3.231 \(\int \frac{9-40 x-18 x^2+174 x^4+24 x^5+26 x^6-39 x^8}{\left (3+2 x^2+x^4\right )^3} \, dx\)

Optimal. Leaf size=59 \[ \frac{13 x}{x^4+2 x^2+3}-\frac{2 \left (13 x^2+18\right ) x}{\left (x^4+2 x^2+3\right )^2}+\frac{2 \left (1-2 x^2\right )}{\left (x^4+2 x^2+3\right )^2} \]

[Out]

(2*(1 - 2*x^2))/(3 + 2*x^2 + x^4)^2 - (2*x*(18 + 13*x^2))/(3 + 2*x^2 + x^4)^2 +
(13*x)/(3 + 2*x^2 + x^4)

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Rubi [A]  time = 0.130675, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.14 \[ \frac{13 x}{x^4+2 x^2+3}-\frac{2 \left (13 x^2+18\right ) x}{\left (x^4+2 x^2+3\right )^2}+\frac{2 \left (1-2 x^2\right )}{\left (x^4+2 x^2+3\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[(9 - 40*x - 18*x^2 + 174*x^4 + 24*x^5 + 26*x^6 - 39*x^8)/(3 + 2*x^2 + x^4)^3,x]

[Out]

(2*(1 - 2*x^2))/(3 + 2*x^2 + x^4)^2 - (2*x*(18 + 13*x^2))/(3 + 2*x^2 + x^4)^2 +
(13*x)/(3 + 2*x^2 + x^4)

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Rubi in Sympy [A]  time = 35.4857, size = 53, normalized size = 0.9 \[ \frac{x \left (- 65536 x^{3} - 131072 x + 3833856\right )}{294912 \left (x^{4} + 2 x^{2} + 3\right )} - \frac{x \left (1024 x^{3} + 39936 x^{2} + 8192 x + 55296\right )}{1536 \left (x^{4} + 2 x^{2} + 3\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-39*x**8+26*x**6+24*x**5+174*x**4-18*x**2-40*x+9)/(x**4+2*x**2+3)**3,x)

[Out]

x*(-65536*x**3 - 131072*x + 3833856)/(294912*(x**4 + 2*x**2 + 3)) - x*(1024*x**3
+ 39936*x**2 + 8192*x + 55296)/(1536*(x**4 + 2*x**2 + 3)**2)

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Mathematica [A]  time = 0.0187123, size = 28, normalized size = 0.47 \[ \frac{13 x^5-4 x^2+3 x+2}{\left (x^4+2 x^2+3\right )^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(9 - 40*x - 18*x^2 + 174*x^4 + 24*x^5 + 26*x^6 - 39*x^8)/(3 + 2*x^2 + x^4)^3,x]

[Out]

(2 + 3*x - 4*x^2 + 13*x^5)/(3 + 2*x^2 + x^4)^2

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Maple [A]  time = 0.01, size = 30, normalized size = 0.5 \[ -{\frac{-13\,{x}^{5}+4\,{x}^{2}-3\,x-2}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-39*x^8+26*x^6+24*x^5+174*x^4-18*x^2-40*x+9)/(x^4+2*x^2+3)^3,x)

[Out]

-(-13*x^5+4*x^2-3*x-2)/(x^4+2*x^2+3)^2

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Maxima [A]  time = 0.854581, size = 51, normalized size = 0.86 \[ \frac{13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(39*x^8 - 26*x^6 - 24*x^5 - 174*x^4 + 18*x^2 + 40*x - 9)/(x^4 + 2*x^2 + 3)^3,x, algorithm="maxima")

[Out]

(13*x^5 - 4*x^2 + 3*x + 2)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)

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Fricas [A]  time = 0.258463, size = 51, normalized size = 0.86 \[ \frac{13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(39*x^8 - 26*x^6 - 24*x^5 - 174*x^4 + 18*x^2 + 40*x - 9)/(x^4 + 2*x^2 + 3)^3,x, algorithm="fricas")

[Out]

(13*x^5 - 4*x^2 + 3*x + 2)/(x^8 + 4*x^6 + 10*x^4 + 12*x^2 + 9)

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Sympy [A]  time = 0.548461, size = 34, normalized size = 0.58 \[ \frac{13 x^{5} - 4 x^{2} + 3 x + 2}{x^{8} + 4 x^{6} + 10 x^{4} + 12 x^{2} + 9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-39*x**8+26*x**6+24*x**5+174*x**4-18*x**2-40*x+9)/(x**4+2*x**2+3)**3,x)

[Out]

(13*x**5 - 4*x**2 + 3*x + 2)/(x**8 + 4*x**6 + 10*x**4 + 12*x**2 + 9)

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GIAC/XCAS [A]  time = 0.262823, size = 38, normalized size = 0.64 \[ \frac{13 \, x^{5} - 4 \, x^{2} + 3 \, x + 2}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(39*x^8 - 26*x^6 - 24*x^5 - 174*x^4 + 18*x^2 + 40*x - 9)/(x^4 + 2*x^2 + 3)^3,x, algorithm="giac")

[Out]

(13*x^5 - 4*x^2 + 3*x + 2)/(x^4 + 2*x^2 + 3)^2