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

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 = 0.433148, 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., Rules used = {} \[ \int \left (\frac{-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3}-\frac{3 \left (1+4 x^3\right ) \left (2-3 x+5 x^2+x^4-5 x^6\right )}{\left (3+x+x^4\right )^4}\right ) \, dx \]

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*D
efer[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]

Rubi steps

\begin{align*} \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 &=-\left (3 \int \frac{\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} \, dx\right )+\int \frac{-3+10 x+4 x^3-30 x^5}{\left (3+x+x^4\right )^3} \, dx\\ &=-\frac{10 x^6}{\left (3+x+x^4\right )^3}+\frac{5 x^2}{\left (3+x+x^4\right )^2}-\frac{1}{6} \int \frac{18+120 x-24 x^3}{\left (3+x+x^4\right )^3} \, dx+\frac{1}{2} \int \frac{-12+18 x-30 x^2-48 x^3+66 x^4+240 x^5+90 x^6-24 x^7}{\left (3+x+x^4\right )^4} \, dx\\ &=\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}-\frac{1}{24} \int \frac{96+480 x}{\left (3+x+x^4\right )^3} \, dx-\frac{1}{16} \int \frac{96-144 x+240 x^2+672 x^3-504 x^4-1920 x^5-720 x^6}{\left (3+x+x^4\right )^4} \, dx\\ &=-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}+\frac{1}{144} \int \frac{-864+1296 x+4320 x^2-6048 x^3+4536 x^4+17280 x^5}{\left (3+x+x^4\right )^4} \, dx-\frac{1}{24} \int \left (\frac{96}{\left (3+x+x^4\right )^3}+\frac{480 x}{\left (3+x+x^4\right )^3}\right ) \, dx\\ &=-\frac{12 x^2}{\left (3+x+x^4\right )^3}-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}-\frac{\int \frac{8640-116640 x-25920 x^2+60480 x^3-45360 x^4}{\left (3+x+x^4\right )^4} \, dx}{1440}-4 \int \frac{1}{\left (3+x+x^4\right )^3} \, dx-20 \int \frac{x}{\left (3+x+x^4\right )^3} \, dx\\ &=-\frac{63 x}{22 \left (3+x+x^4\right )^3}-\frac{12 x^2}{\left (3+x+x^4\right )^3}-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}+\frac{\int \frac{41040+1192320 x+285120 x^2-665280 x^3}{\left (3+x+x^4\right )^4} \, dx}{15840}-4 \int \frac{1}{\left (3+x+x^4\right )^3} \, dx-20 \int \frac{x}{\left (3+x+x^4\right )^3} \, dx\\ &=\frac{7}{2 \left (3+x+x^4\right )^3}-\frac{63 x}{22 \left (3+x+x^4\right )^3}-\frac{12 x^2}{\left (3+x+x^4\right )^3}-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}+\frac{\int \frac{829440+4769280 x+1140480 x^2}{\left (3+x+x^4\right )^4} \, dx}{63360}-4 \int \frac{1}{\left (3+x+x^4\right )^3} \, dx-20 \int \frac{x}{\left (3+x+x^4\right )^3} \, dx\\ &=\frac{7}{2 \left (3+x+x^4\right )^3}-\frac{63 x}{22 \left (3+x+x^4\right )^3}-\frac{12 x^2}{\left (3+x+x^4\right )^3}-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}+\frac{\int \left (\frac{829440}{\left (3+x+x^4\right )^4}+\frac{4769280 x}{\left (3+x+x^4\right )^4}+\frac{1140480 x^2}{\left (3+x+x^4\right )^4}\right ) \, dx}{63360}-4 \int \frac{1}{\left (3+x+x^4\right )^3} \, dx-20 \int \frac{x}{\left (3+x+x^4\right )^3} \, dx\\ &=\frac{7}{2 \left (3+x+x^4\right )^3}-\frac{63 x}{22 \left (3+x+x^4\right )^3}-\frac{12 x^2}{\left (3+x+x^4\right )^3}-\frac{5 x^3}{\left (3+x+x^4\right )^3}+\frac{3 x^4}{2 \left (3+x+x^4\right )^3}-\frac{10 x^6}{\left (3+x+x^4\right )^3}-\frac{1}{2 \left (3+x+x^4\right )^2}+\frac{5 x^2}{\left (3+x+x^4\right )^2}-4 \int \frac{1}{\left (3+x+x^4\right )^3} \, dx+\frac{144}{11} \int \frac{1}{\left (3+x+x^4\right )^4} \, dx+18 \int \frac{x^2}{\left (3+x+x^4\right )^4} \, dx-20 \int \frac{x}{\left (3+x+x^4\right )^3} \, dx+\frac{828}{11} \int \frac{x}{\left (3+x+x^4\right )^4} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0105844, 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 = 0.016, size = 112, normalized size = 4.2 \begin{align*} -{\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 ) } \end{align*}

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^4-68792/195075*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^8-988
24/585225*x^7-1322894/585225*x^6+36022/585225*x^5-129019/1170450*x^4-790303/585225*x^3-80674/65025*x^2-10951/1
4450*x+26831/43350)/(x^4+x+3)^3

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Maxima [B]  time = 1.31228, size = 88, normalized size = 3.26 \begin{align*} -\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} \end{align*}

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, algorithm="ma
xima")

[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 [B]  time = 0.940147, size = 147, normalized size = 5.44 \begin{align*} -\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} \end{align*}

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, algorithm="fr
icas")

[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 [B]  time = 0.285799, size = 61, normalized size = 2.26 \begin{align*} - \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} \end{align*}

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 [B]  time = 1.1459, size = 150, normalized size = 5.56 \begin{align*} \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}} \end{align*}

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, algorithm="gi
ac")

[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 + 1580606*x^3 + 1452132*x^2 + 887031*x - 724437)/(x^4 + x + 3)^3