3.492 \(\int -\frac{15-36 x+5 x^2+12 x^3-34 x^4+140 x^5+15 x^6+8 x^7-30 x^9}{\left (3+x+x^4\right )^4} \, dx\)

Optimal. Leaf size=60 \[ \frac{x^4}{\left (x^4+x+3\right )^3}-\frac{3 x}{\left (x^4+x+3\right )^3}+\frac{2}{\left (x^4+x+3\right )^3}-\frac{5 x^6}{\left (x^4+x+3\right )^3}+\frac{5 x^2}{\left (x^4+x+3\right )^3} \]

[Out]

2/(3 + x + x^4)^3 - (3*x)/(3 + x + x^4)^3 + (5*x^2)/(3 + x + x^4)^3 + x^4/(3 + x
+ x^4)^3 - (5*x^6)/(3 + x + x^4)^3

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Rubi [A]  time = 0.231907, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ \frac{x^4}{\left (x^4+x+3\right )^3}-\frac{3 x}{\left (x^4+x+3\right )^3}+\frac{2}{\left (x^4+x+3\right )^3}-\frac{5 x^6}{\left (x^4+x+3\right )^3}+\frac{5 x^2}{\left (x^4+x+3\right )^3} \]

Antiderivative was successfully verified.

[In]  Int[-((15 - 36*x + 5*x^2 + 12*x^3 - 34*x^4 + 140*x^5 + 15*x^6 + 8*x^7 - 30*x^9)/(3 + x + x^4)^4),x]

[Out]

2/(3 + x + x^4)^3 - (3*x)/(3 + x + x^4)^3 + (5*x^2)/(3 + x + x^4)^3 + x^4/(3 + x
+ x^4)^3 - (5*x^6)/(3 + x + x^4)^3

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Rubi in Sympy [A]  time = 57.8957, size = 60, normalized size = 1. \[ - \frac{5 x^{6}}{\left (x^{4} + x + 3\right )^{3}} + \frac{x^{4}}{\left (x^{4} + x + 3\right )^{3}} + \frac{5 x^{2}}{\left (x^{4} + x + 3\right )^{3}} - \frac{3 x}{\left (x^{4} + x + 3\right )^{3}} + \frac{2}{\left (x^{4} + x + 3\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((30*x**9-8*x**7-15*x**6-140*x**5+34*x**4-12*x**3-5*x**2+36*x-15)/(x**4+x+3)**4,x)

[Out]

-5*x**6/(x**4 + x + 3)**3 + x**4/(x**4 + x + 3)**3 + 5*x**2/(x**4 + x + 3)**3 -
3*x/(x**4 + x + 3)**3 + 2/(x**4 + x + 3)**3

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Mathematica [A]  time = 0.0232864, size = 27, normalized size = 0.45 \[ \frac{-5 x^6+x^4+5 x^2-3 x+2}{\left (x^4+x+3\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[-((15 - 36*x + 5*x^2 + 12*x^3 - 34*x^4 + 140*x^5 + 15*x^6 + 8*x^7 - 30*x^9)/(3 + x + x^4)^4),x]

[Out]

(2 - 3*x + 5*x^2 + x^4 - 5*x^6)/(3 + x + x^4)^3

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Maple [A]  time = 0.012, size = 28, normalized size = 0.5 \[{\frac{-5\,{x}^{6}+{x}^{4}+5\,{x}^{2}-3\,x+2}{ \left ({x}^{4}+x+3 \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((30*x^9-8*x^7-15*x^6-140*x^5+34*x^4-12*x^3-5*x^2+36*x-15)/(x^4+x+3)^4,x)

[Out]

(-5*x^6+x^4+5*x^2-3*x+2)/(x^4+x+3)^3

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Maxima [A]  time = 0.798956, size = 88, normalized size = 1.47 \[ -\frac{5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((30*x^9 - 8*x^7 - 15*x^6 - 140*x^5 + 34*x^4 - 12*x^3 - 5*x^2 + 36*x - 15)/(x^4 + x + 3)^4,x, algorithm="maxima")

[Out]

-(5*x^6 - x^4 - 5*x^2 + 3*x - 2)/(x^12 + 3*x^9 + 9*x^8 + 3*x^6 + 18*x^5 + 27*x^4
+ x^3 + 9*x^2 + 27*x + 27)

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Fricas [A]  time = 0.255867, size = 88, normalized size = 1.47 \[ -\frac{5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((30*x^9 - 8*x^7 - 15*x^6 - 140*x^5 + 34*x^4 - 12*x^3 - 5*x^2 + 36*x - 15)/(x^4 + x + 3)^4,x, algorithm="fricas")

[Out]

-(5*x^6 - x^4 - 5*x^2 + 3*x - 2)/(x^12 + 3*x^9 + 9*x^8 + 3*x^6 + 18*x^5 + 27*x^4
+ x^3 + 9*x^2 + 27*x + 27)

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Sympy [A]  time = 0.823337, size = 61, normalized size = 1.02 \[ - \frac{5 x^{6} - x^{4} - 5 x^{2} + 3 x - 2}{x^{12} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 18 x^{5} + 27 x^{4} + x^{3} + 9 x^{2} + 27 x + 27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((30*x**9-8*x**7-15*x**6-140*x**5+34*x**4-12*x**3-5*x**2+36*x-15)/(x**4+x+3)**4,x)

[Out]

-(5*x**6 - x**4 - 5*x**2 + 3*x - 2)/(x**12 + 3*x**9 + 9*x**8 + 3*x**6 + 18*x**5
+ 27*x**4 + x**3 + 9*x**2 + 27*x + 27)

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GIAC/XCAS [A]  time = 0.259847, size = 41, normalized size = 0.68 \[ -\frac{5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{{\left (x^{4} + x + 3\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((30*x^9 - 8*x^7 - 15*x^6 - 140*x^5 + 34*x^4 - 12*x^3 - 5*x^2 + 36*x - 15)/(x^4 + x + 3)^4,x, algorithm="giac")

[Out]

-(5*x^6 - x^4 - 5*x^2 + 3*x - 2)/(x^4 + x + 3)^3