\(\int (\frac {3 (-47+228 x+120 x^2+19 x^3)}{(3+x+x^4)^4}+\frac {42-320 x-75 x^2-8 x^3}{(3+x+x^4)^3}+\frac {30 x}{(3+x+x^4)^2}) \, dx\) [493]

Optimal result

Integrand size = 61, antiderivative size = 27 \[ \int \left (\frac {3 \left (-47+228 x+120 x^2+19 x^3\right )}{\left (3+x+x^4\right )^4}+\frac {42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3}+\frac {30 x}{\left (3+x+x^4\right )^2}\right ) \, dx=\frac {2-3 x+5 x^2+x^4-5 x^6}{\left (3+x+x^4\right )^3} \]

[Out]

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

Rubi [F]

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

[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*} \text {integral}& = 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] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \left (\frac {3 \left (-47+228 x+120 x^2+19 x^3\right )}{\left (3+x+x^4\right )^4}+\frac {42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3}+\frac {30 x}{\left (3+x+x^4\right )^2}\right ) \, dx=\frac {2-3 x+5 x^2+x^4-5 x^6}{\left (3+x+x^4\right )^3} \]

[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

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04

method	result	size
norman	\(\frac {-5 x^{6}+x^{4}+5 x^{2}-3 x +2}{\left (x^{4}+x +3\right )^{3}}\)	\(28\)
parallelrisch	\(\frac {-5 x^{6}+x^{4}+5 x^{2}-3 x +2}{\left (x^{4}+x +3\right )^{3}}\)	\(28\)
gosper	\(-\frac {5 x^{6}-x^{4}-5 x^{2}+3 x -2}{\left (x^{4}+x +3\right )^{3}}\)	\(31\)
default	\(\frac {\frac {377432}{195075} x^{7}-\frac {1404328}{195075} x^{6}+\frac {234517}{195075} x^{5}+\frac {660506}{195075} x^{4}-\frac {208792}{195075} x^{3}-\frac {13339729}{390150} x^{2}+\frac {89881}{13005} x +\frac {121303}{21675}}{\left (x^{4}+x +3\right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (377432 \textit {\_R}^{2}-2808656 \textit {\_R} +703551\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{195075}+\frac {-\frac {32}{51} x^{3}+\frac {128}{51} x^{2}-\frac {2}{51} x -\frac {8}{17}}{x^{4}+x +3}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (-16 \textit {\_R}^{2}+128 \textit {\_R} -3\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{51}+\frac {-\frac {255032}{195075} x^{11}+\frac {914728}{195075} x^{10}-\frac {226867}{195075} x^{9}-\frac {701338}{195075} x^{8}+\frac {236024}{195075} x^{7}+\frac {13501313}{390150} x^{6}-\frac {2360372}{195075} x^{5}-\frac {1873778}{195075} x^{4}+\frac {10935781}{390150} x^{3}+\frac {3415123}{43350} x^{2}-\frac {188961}{7225} x -\frac {76253}{7225}}{\left (x^{4}+x +3\right )^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (-255032 \textit {\_R}^{2}+1829456 \textit {\_R} -680601\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{195075}\)	\(250\)
risch	\(\frac {\frac {377432}{195075} x^{7}-\frac {1404328}{195075} x^{6}+\frac {234517}{195075} x^{5}+\frac {660506}{195075} x^{4}-\frac {208792}{195075} x^{3}-\frac {13339729}{390150} x^{2}+\frac {89881}{13005} x +\frac {121303}{21675}}{\left (x^{4}+x +3\right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (377432 \textit {\_R}^{2}-2808656 \textit {\_R} +703551\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{195075}+\frac {-\frac {32}{51} x^{3}+\frac {128}{51} x^{2}-\frac {2}{51} x -\frac {8}{17}}{x^{4}+x +3}+\frac {2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (-16 \textit {\_R}^{2}+128 \textit {\_R} -3\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{51}+\frac {-\frac {255032}{195075} x^{11}+\frac {914728}{195075} x^{10}-\frac {226867}{195075} x^{9}-\frac {701338}{195075} x^{8}+\frac {236024}{195075} x^{7}+\frac {13501313}{390150} x^{6}-\frac {2360372}{195075} x^{5}-\frac {1873778}{195075} x^{4}+\frac {10935781}{390150} x^{3}+\frac {3415123}{43350} x^{2}-\frac {188961}{7225} x -\frac {76253}{7225}}{\left (x^{4}+x +3\right )^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )}{\sum }\frac {\left (-255032 \textit {\_R}^{2}+1829456 \textit {\_R} -680601\right ) \ln \left (x -\textit {\_R} \right )}{4 \textit {\_R}^{3}+1}\right )}{195075}\)	\(250\)

[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,method=_RE
TURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (30) = 60\).

Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \left (\frac {3 \left (-47+228 x+120 x^2+19 x^3\right )}{\left (3+x+x^4\right )^4}+\frac {42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3}+\frac {30 x}{\left (3+x+x^4\right )^2}\right ) \, dx=-\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (26) = 52\).

Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.22 \[ \int \left (\frac {3 \left (-47+228 x+120 x^2+19 x^3\right )}{\left (3+x+x^4\right )^4}+\frac {42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3}+\frac {30 x}{\left (3+x+x^4\right )^2}\right ) \, dx=\frac {- 5 x^{6} + x^{4} + 5 x^{2} - 3 x + 2}{x^{12} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 18 x^{5} + 27 x^{4} + x^{3} + 9 x^{2} + 27 x + 27} \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (30) = 60\).

Time = 0.23 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.41 \[ \int \left (\frac {3 \left (-47+228 x+120 x^2+19 x^3\right )}{\left (3+x+x^4\right )^4}+\frac {42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3}+\frac {30 x}{\left (3+x+x^4\right )^2}\right ) \, dx=-\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]

[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)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (30) = 60\).

Time = 0.28 (sec) , antiderivative size = 197, normalized size of antiderivative = 7.30 \[ \int \left (\frac {3 \left (-47+228 x+120 x^2+19 x^3\right )}{\left (3+x+x^4\right )^4}+\frac {42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3}+\frac {30 x}{\left (3+x+x^4\right )^2}\right ) \, dx=\frac {1}{195075} \, x {\left (\frac {377432 \, x^{2} - 2808656 \, x + 703551}{x^{4} + x + 3} - \frac {255032 \, x^{2} - 1829456 \, x + 680601}{x^{4} + x + 3} - \frac {7650 \, {\left (16 \, x^{2} - 128 \, x + 3\right )}}{x^{4} + x + 3}\right )} - \frac {2 \, {\left (16 \, x^{3} - 64 \, x^{2} + x + 12\right )}}{51 \, {\left (x^{4} + x + 3\right )}} + \frac {754864 \, x^{7} - 2808656 \, x^{6} + 469034 \, x^{5} + 1321012 \, x^{4} - 417584 \, x^{3} - 13339729 \, x^{2} + 2696430 \, x + 2183454}{390150 \, {\left (x^{4} + x + 3\right )}^{2}} - \frac {510064 \, x^{11} - 1829456 \, x^{10} + 453734 \, x^{9} + 1402676 \, x^{8} - 472048 \, x^{7} - 13501313 \, x^{6} + 4720744 \, x^{5} + 3747556 \, x^{4} - 10935781 \, x^{3} - 30736107 \, x^{2} + 10203894 \, x + 4117662}{390150 \, {\left (x^{4} + x + 3\right )}^{3}} \]

[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]

1/195075*x*((377432*x^2 - 2808656*x + 703551)/(x^4 + x + 3) - (255032*x^2 - 1829456*x + 680601)/(x^4 + x + 3)
- 7650*(16*x^2 - 128*x + 3)/(x^4 + x + 3)) - 2/51*(16*x^3 - 64*x^2 + x + 12)/(x^4 + x + 3) + 1/390150*(754864*
x^7 - 2808656*x^6 + 469034*x^5 + 1321012*x^4 - 417584*x^3 - 13339729*x^2 + 2696430*x + 2183454)/(x^4 + x + 3)^
2 - 1/390150*(510064*x^11 - 1829456*x^10 + 453734*x^9 + 1402676*x^8 - 472048*x^7 - 13501313*x^6 + 4720744*x^5
+ 3747556*x^4 - 10935781*x^3 - 30736107*x^2 + 10203894*x + 4117662)/(x^4 + x + 3)^3

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \left (\frac {3 \left (-47+228 x+120 x^2+19 x^3\right )}{\left (3+x+x^4\right )^4}+\frac {42-320 x-75 x^2-8 x^3}{\left (3+x+x^4\right )^3}+\frac {30 x}{\left (3+x+x^4\right )^2}\right ) \, dx=\frac {-5\,x^6+x^4+5\,x^2-3\,x+2}{{\left (x^4+x+3\right )}^3} \]

[In]

int((684*x + 360*x^2 + 57*x^3 - 141)/(x + x^4 + 3)^4 - (320*x + 75*x^2 + 8*x^3 - 42)/(x + x^4 + 3)^3 + (30*x)/
(x + x^4 + 3)^2,x)

[Out]

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