3.493 \(\int (\frac{3 (-47+228 x+120 x^2+19 x^3)}{(3+x+x^4)^4}+\frac{42-320 x-75 x^2-8 x^3}{(3+x+x^4)^3}+\frac{30 x}{(3+x+x^4)^2}) \, dx\)

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 = 0.311277, 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 = {} \[ \int \left (\frac{3 \left (-47+228 x+120 x^2+19 x^3\right )}{\left (3+x+x^4\right )^4}+\frac{42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3}+\frac{30 x}{\left (3+x+x^4\right )^2}\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(3*(-47 + 228*x + 120*x^2 + 19*x^3))/(3 + x + x^4)^4 + (42 - 320*x - 75*x^2 - 8*x^3)/(3 + x + x^4)^3 + (30
*x)/(3 + x + x^4)^2,x]

[Out]

-19/(4*(3 + x + x^4)^3) + (3 + x + x^4)^(-2) - (621*Defer[Int][(3 + x + x^4)^(-4), x])/4 + 684*Defer[Int][x/(3
+ x + x^4)^4, x] + 360*Defer[Int][x^2/(3 + x + x^4)^4, x] + 44*Defer[Int][(3 + x + x^4)^(-3), x] - 320*Defer[
Int][x/(3 + x + x^4)^3, x] - 75*Defer[Int][x^2/(3 + x + x^4)^3, x] + 30*Defer[Int][x/(3 + x + x^4)^2, x]

Rubi steps

\begin{align*} \int \left (\frac{3 \left (-47+228 x+120 x^2+19 x^3\right )}{\left (3+x+x^4\right )^4}+\frac{42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3}+\frac{30 x}{\left (3+x+x^4\right )^2}\right ) \, dx &=3 \int \frac{-47+228 x+120 x^2+19 x^3}{\left (3+x+x^4\right )^4} \, dx+30 \int \frac{x}{\left (3+x+x^4\right )^2} \, dx+\int \frac{42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3} \, dx\\ &=-\frac{19}{4 \left (3+x+x^4\right )^3}+\frac{1}{\left (3+x+x^4\right )^2}+\frac{1}{4} \int \frac{176-1280 x-300 x^2}{\left (3+x+x^4\right )^3} \, dx+\frac{3}{4} \int \frac{-207+912 x+480 x^2}{\left (3+x+x^4\right )^4} \, dx+30 \int \frac{x}{\left (3+x+x^4\right )^2} \, dx\\ &=-\frac{19}{4 \left (3+x+x^4\right )^3}+\frac{1}{\left (3+x+x^4\right )^2}+\frac{1}{4} \int \left (\frac{176}{\left (3+x+x^4\right )^3}-\frac{1280 x}{\left (3+x+x^4\right )^3}-\frac{300 x^2}{\left (3+x+x^4\right )^3}\right ) \, dx+\frac{3}{4} \int \left (-\frac{207}{\left (3+x+x^4\right )^4}+\frac{912 x}{\left (3+x+x^4\right )^4}+\frac{480 x^2}{\left (3+x+x^4\right )^4}\right ) \, dx+30 \int \frac{x}{\left (3+x+x^4\right )^2} \, dx\\ &=-\frac{19}{4 \left (3+x+x^4\right )^3}+\frac{1}{\left (3+x+x^4\right )^2}+30 \int \frac{x}{\left (3+x+x^4\right )^2} \, dx+44 \int \frac{1}{\left (3+x+x^4\right )^3} \, dx-75 \int \frac{x^2}{\left (3+x+x^4\right )^3} \, dx-\frac{621}{4} \int \frac{1}{\left (3+x+x^4\right )^4} \, dx-320 \int \frac{x}{\left (3+x+x^4\right )^3} \, dx+360 \int \frac{x^2}{\left (3+x+x^4\right )^4} \, dx+684 \int \frac{x}{\left (3+x+x^4\right )^4} \, dx\\ \end{align*}

Mathematica [A]  time = 0.0094752, 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*(-47 + 228*x + 120*x^2 + 19*x^3))/(3 + x + x^4)^4 + (42 - 320*x - 75*x^2 - 8*x^3)/(3 + x + x^4)^3
+ (30*x)/(3 + x + x^4)^2,x]

[Out]

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

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Maple [C]  time = 0.024, size = 250, normalized size = 9.3 \begin{align*}{\frac{1}{ \left ({x}^{4}+x+3 \right ) ^{2}} \left ({\frac{377432\,{x}^{7}}{195075}}-{\frac{1404328\,{x}^{6}}{195075}}+{\frac{234517\,{x}^{5}}{195075}}+{\frac{660506\,{x}^{4}}{195075}}-{\frac{208792\,{x}^{3}}{195075}}-{\frac{13339729\,{x}^{2}}{390150}}+{\frac{89881\,x}{13005}}+{\frac{121303}{21675}} \right ) }+{\frac{1}{195075}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}+{\it \_Z}+3 \right ) }{\frac{ \left ( 377432\,{{\it \_R}}^{2}-2808656\,{\it \_R}+703551 \right ) \ln \left ( x-{\it \_R} \right ) }{4\,{{\it \_R}}^{3}+1}}}+30\,{\frac{1}{{x}^{4}+x+3} \left ( -{\frac{16\,{x}^{3}}{765}}+{\frac{64\,{x}^{2}}{765}}-{\frac{x}{765}}-{\frac{4}{255}} \right ) }+{\frac{2}{51}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}+{\it \_Z}+3 \right ) }{\frac{ \left ( -16\,{{\it \_R}}^{2}+128\,{\it \_R}-3 \right ) \ln \left ( x-{\it \_R} \right ) }{4\,{{\it \_R}}^{3}+1}}}+3\,{\frac{1}{ \left ({x}^{4}+x+3 \right ) ^{3}} \left ( -{\frac{255032\,{x}^{11}}{585225}}+{\frac{914728\,{x}^{10}}{585225}}-{\frac{226867\,{x}^{9}}{585225}}-{\frac{701338\,{x}^{8}}{585225}}+{\frac{236024\,{x}^{7}}{585225}}+{\frac{13501313\,{x}^{6}}{1170450}}-{\frac{2360372\,{x}^{5}}{585225}}-{\frac{1873778\,{x}^{4}}{585225}}+{\frac{10935781\,{x}^{3}}{1170450}}+{\frac{3415123\,{x}^{2}}{130050}}-{\frac{62987\,x}{7225}}-{\frac{76253}{21675}} \right ) }+{\frac{1}{195075}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{4}+{\it \_Z}+3 \right ) }{\frac{ \left ( -255032\,{{\it \_R}}^{2}+1829456\,{\it \_R}-680601 \right ) \ln \left ( x-{\it \_R} \right ) }{4\,{{\it \_R}}^{3}+1}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(3*(19*x^3+120*x^2+228*x-47)/(x^4+x+3)^4+(-8*x^3-75*x^2-320*x+42)/(x^4+x+3)^3+30*x/(x^4+x+3)^2,x)

[Out]

(377432/195075*x^7-1404328/195075*x^6+234517/195075*x^5+660506/195075*x^4-208792/195075*x^3-13339729/390150*x^
2+89881/13005*x+121303/21675)/(x^4+x+3)^2+1/195075*sum((377432*_R^2-2808656*_R+703551)/(4*_R^3+1)*ln(x-_R),_R=
RootOf(_Z^4+_Z+3))+30*(-16/765*x^3+64/765*x^2-1/765*x-4/255)/(x^4+x+3)+2/51*sum((-16*_R^2+128*_R-3)/(4*_R^3+1)
*ln(x-_R),_R=RootOf(_Z^4+_Z+3))+3*(-255032/585225*x^11+914728/585225*x^10-226867/585225*x^9-701338/585225*x^8+
236024/585225*x^7+13501313/1170450*x^6-2360372/585225*x^5-1873778/585225*x^4+10935781/1170450*x^3+3415123/1300
50*x^2-62987/7225*x-76253/21675)/(x^4+x+3)^3+1/195075*sum((-255032*_R^2+1829456*_R-680601)/(4*_R^3+1)*ln(x-_R)
,_R=RootOf(_Z^4+_Z+3))

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Maxima [B]  time = 1.32357, 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(3*(19*x^3+120*x^2+228*x-47)/(x^4+x+3)^4+(-8*x^3-75*x^2-320*x+42)/(x^4+x+3)^3+30*x/(x^4+x+3)^2,x, alg
orithm="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 [B]  time = 0.922794, 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(3*(19*x^3+120*x^2+228*x-47)/(x^4+x+3)^4+(-8*x^3-75*x^2-320*x+42)/(x^4+x+3)^3+30*x/(x^4+x+3)^2,x, alg
orithm="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 [B]  time = 0.317374, 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(3*(19*x**3+120*x**2+228*x-47)/(x**4+x+3)**4+(-8*x**3-75*x**2-320*x+42)/(x**4+x+3)**3+30*x/(x**4+x+3)
**2,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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{30 \, x}{{\left (x^{4} + x + 3\right )}^{2}} - \frac{8 \, x^{3} + 75 \, x^{2} + 320 \, x - 42}{{\left (x^{4} + x + 3\right )}^{3}} + \frac{3 \,{\left (19 \, x^{3} + 120 \, x^{2} + 228 \, x - 47\right )}}{{\left (x^{4} + x + 3\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(3*(19*x^3+120*x^2+228*x-47)/(x^4+x+3)^4+(-8*x^3-75*x^2-320*x+42)/(x^4+x+3)^3+30*x/(x^4+x+3)^2,x, alg
orithm="giac")

[Out]

integrate(30*x/(x^4 + x + 3)^2 - (8*x^3 + 75*x^2 + 320*x - 42)/(x^4 + x + 3)^3 + 3*(19*x^3 + 120*x^2 + 228*x -
47)/(x^4 + x + 3)^4, x)