3.494 \(\int \left (\frac{-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac{3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx\)

Optimal. Leaf size=27 \[ \frac{-5 x^6+x^4+5 x^2-3 x+2}{\left (x^4+x+3\right )^3} \]

[Out]

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

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Rubi [F]  time = 0.786527, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ \text{Int}\left (\frac{-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac{3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4},x\right ) \]

Verification is Not applicable to the result.

[In]  Int[(-3 + 10*x + 4*x^3 - 30*x^5)/(3 + x + x^4)^3 - (3*(1 + 4*x^3)*(2 - 3*x + 5*x^2 + x^4 - 5*x^6))/(3 + x + x^4)^4,x]

[Out]

7/(2*(3 + x + x^4)^3) - (63*x)/(22*(3 + x + x^4)^3) - (12*x^2)/(3 + x + x^4)^3 -
(5*x^3)/(3 + x + x^4)^3 + (3*x^4)/(2*(3 + x + x^4)^3) - (10*x^6)/(3 + x + x^4)^
3 - 1/(2*(3 + x + x^4)^2) + (5*x^2)/(3 + x + x^4)^2 + (144*Defer[Int][(3 + x + x
^4)^(-4), x])/11 + (828*Defer[Int][x/(3 + x + x^4)^4, x])/11 + 18*Defer[Int][x^2
/(3 + x + x^4)^4, x] - 4*Defer[Int][(3 + x + x^4)^(-3), x] - 20*Defer[Int][x/(3
+ x + x^4)^3, x]

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-30*x**5+4*x**3+10*x-3)/(x**4+x+3)**3-3*(4*x**3+1)*(-5*x**6+x**4+5*x**2-3*x+2)/(x**4+x+3)**4,x)

[Out]

Timed out

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Mathematica [A]  time = 0.0138268, size = 27, normalized size = 1. \[ \frac{-5 x^6+x^4+5 x^2-3 x+2}{\left (x^4+x+3\right )^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(-3 + 10*x + 4*x^3 - 30*x^5)/(3 + x + x^4)^3 - (3*(1 + 4*x^3)*(2 - 3*x + 5*x^2 + x^4 - 5*x^6))/(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 [B]  time = 0.024, size = 112, normalized size = 4.2 \[ -{\frac{1}{ \left ({x}^{4}+x+3 \right ) ^{2}} \left ( -{\frac{34568\,{x}^{7}}{195075}}+{\frac{73672\,{x}^{6}}{195075}}+{\frac{15392\,{x}^{5}}{195075}}-{\frac{60494\,{x}^{4}}{195075}}-{\frac{68792\,{x}^{3}}{195075}}-{\frac{583927\,{x}^{2}}{195075}}+{\frac{3356\,x}{13005}}-{\frac{2069}{43350}} \right ) }+3\,{\frac{1}{ \left ({x}^{4}+x+3 \right ) ^{3}} \left ( -{\frac{34568\,{x}^{11}}{585225}}+{\frac{73672\,{x}^{10}}{585225}}+{\frac{15392\,{x}^{9}}{585225}}-{\frac{95062\,{x}^{8}}{585225}}-{\frac{98824\,{x}^{7}}{585225}}-{\frac{1322894\,{x}^{6}}{585225}}+{\frac{36022\,{x}^{5}}{585225}}-{\frac{129019\,{x}^{4}}{1170450}}-{\frac{790303\,{x}^{3}}{585225}}-{\frac{80674\,{x}^{2}}{65025}}-{\frac{10951\,x}{14450}}+{\frac{26831}{43350}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-30*x^5+4*x^3+10*x-3)/(x^4+x+3)^3-3*(4*x^3+1)*(-5*x^6+x^4+5*x^2-3*x+2)/(x^4+x+3)^4,x)

[Out]

-(-34568/195075*x^7+73672/195075*x^6+15392/195075*x^5-60494/195075*x^4-68792/195
075*x^3-583927/195075*x^2+3356/13005*x-2069/43350)/(x^4+x+3)^2+3*(-34568/585225*
x^11+73672/585225*x^10+15392/585225*x^9-95062/585225*x^8-98824/585225*x^7-132289
4/585225*x^6+36022/585225*x^5-129019/1170450*x^4-790303/585225*x^3-80674/65025*x
^2-10951/14450*x+26831/43350)/(x^4+x+3)^3

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Maxima [A]  time = 0.810754, size = 88, normalized size = 3.26 \[ -\frac{5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(30*x^5 - 4*x^3 - 10*x + 3)/(x^4 + x + 3)^3 + 3*(5*x^6 - x^4 - 5*x^2 + 3*x - 2)*(4*x^3 + 1)/(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 = 0.277168, size = 88, normalized size = 3.26 \[ -\frac{5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(30*x^5 - 4*x^3 - 10*x + 3)/(x^4 + x + 3)^3 + 3*(5*x^6 - x^4 - 5*x^2 + 3*x - 2)*(4*x^3 + 1)/(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 = 0.952438, size = 61, normalized size = 2.26 \[ - \frac{5 x^{6} - x^{4} - 5 x^{2} + 3 x - 2}{x^{12} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 18 x^{5} + 27 x^{4} + x^{3} + 9 x^{2} + 27 x + 27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-30*x**5+4*x**3+10*x-3)/(x**4+x+3)**3-3*(4*x**3+1)*(-5*x**6+x**4+5*x**2-3*x+2)/(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 = 0.26246, size = 150, normalized size = 5.56 \[ \frac{69136 \, x^{7} - 147344 \, x^{6} - 30784 \, x^{5} + 120988 \, x^{4} + 137584 \, x^{3} + 1167854 \, x^{2} - 100680 \, x + 18621}{390150 \,{\left (x^{4} + x + 3\right )}^{2}} - \frac{69136 \, x^{11} - 147344 \, x^{10} - 30784 \, x^{9} + 190124 \, x^{8} + 197648 \, x^{7} + 2645788 \, x^{6} - 72044 \, x^{5} + 129019 \, x^{4} + 1580606 \, x^{3} + 1452132 \, x^{2} + 887031 \, x - 724437}{390150 \,{\left (x^{4} + x + 3\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(30*x^5 - 4*x^3 - 10*x + 3)/(x^4 + x + 3)^3 + 3*(5*x^6 - x^4 - 5*x^2 + 3*x - 2)*(4*x^3 + 1)/(x^4 + x + 3)^4,x, algorithm="giac")

[Out]

1/390150*(69136*x^7 - 147344*x^6 - 30784*x^5 + 120988*x^4 + 137584*x^3 + 1167854
*x^2 - 100680*x + 18621)/(x^4 + x + 3)^2 - 1/390150*(69136*x^11 - 147344*x^10 -
30784*x^9 + 190124*x^8 + 197648*x^7 + 2645788*x^6 - 72044*x^5 + 129019*x^4 + 158
0606*x^3 + 1452132*x^2 + 887031*x - 724437)/(x^4 + x + 3)^3