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]

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

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Rubi [F]  time = 0.31, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [A]  time = 0.01, size = 27, normalized size = 1.00 \[ \frac {-5 x^6+x^4+5 x^2-3 x+2}{\left (x^4+x+3\right )^3} \]

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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fricas [B]  time = 0.68, size = 65, normalized size = 2.41 \[ -\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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giac [B]  time = 0.40, size = 197, normalized size = 7.30 \[ \frac {1}{195075} \, x {\left (\frac {377432 \, x^{2} - 2808656 \, x + 703551}{x^{4} + x + 3} - \frac {255032 \, x^{2} - 1829456 \, x + 680601}{x^{4} + x + 3} - \frac {7650 \, {\left (16 \, x^{2} - 128 \, x + 3\right )}}{x^{4} + x + 3}\right )} - \frac {2 \, {\left (16 \, x^{3} - 64 \, x^{2} + x + 12\right )}}{51 \, {\left (x^{4} + x + 3\right )}} + \frac {754864 \, x^{7} - 2808656 \, x^{6} + 469034 \, x^{5} + 1321012 \, x^{4} - 417584 \, x^{3} - 13339729 \, x^{2} + 2696430 \, x + 2183454}{390150 \, {\left (x^{4} + x + 3\right )}^{2}} - \frac {510064 \, x^{11} - 1829456 \, x^{10} + 453734 \, x^{9} + 1402676 \, x^{8} - 472048 \, x^{7} - 13501313 \, x^{6} + 4720744 \, x^{5} + 3747556 \, x^{4} - 10935781 \, x^{3} - 30736107 \, x^{2} + 10203894 \, x + 4117662}{390150 \, {\left (x^{4} + x + 3\right )}^{3}} \]

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]

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

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maple [C]  time = 0.03, size = 250, normalized size = 9.26 \[ \frac {\left (-255032 \RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )^{2}+1829456 \RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )-680601\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )+x \right )}{780300 \RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )^{3}+195075}+\frac {2 \left (-16 \RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )^{2}+128 \RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )-3\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )+x \right )}{51 \left (4 \RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )^{3}+1\right )}+\frac {\left (377432 \RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )^{2}-2808656 \RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )+703551\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )+x \right )}{780300 \RootOf \left (\textit {\_Z}^{4}+\textit {\_Z} +3\right )^{3}+195075}+\frac {\frac {377432}{195075} x^{7}-\frac {1404328}{195075} x^{6}+\frac {234517}{195075} x^{5}+\frac {660506}{195075} x^{4}-\frac {208792}{195075} x^{3}-\frac {13339729}{390150} x^{2}+\frac {89881}{13005} x +\frac {121303}{21675}}{\left (x^{4}+x +3\right )^{2}}+\frac {-\frac {32}{51} x^{3}+\frac {128}{51} x^{2}-\frac {2}{51} x -\frac {8}{17}}{x^{4}+x +3}+\frac {-\frac {255032}{195075} x^{11}+\frac {914728}{195075} x^{10}-\frac {226867}{195075} x^{9}-\frac {701338}{195075} x^{8}+\frac {236024}{195075} x^{7}+\frac {13501313}{390150} x^{6}-\frac {2360372}{195075} x^{5}-\frac {1873778}{195075} x^{4}+\frac {10935781}{390150} x^{3}+\frac {3415123}{43350} x^{2}-\frac {188961}{7225} x -\frac {76253}{7225}}{\left (x^{4}+x +3\right )^{3}} \]

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(-_R+x),_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(-_R+x),_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+3415123/13
0050*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(-_R
+x),_R=RootOf(_Z^4+_Z+3))

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maxima [B]  time = 1.20, size = 65, normalized size = 2.41 \[ -\frac {5 \, x^{6} - x^{4} - 5 \, x^{2} + 3 \, x - 2}{x^{12} + 3 \, x^{9} + 9 \, x^{8} + 3 \, x^{6} + 18 \, x^{5} + 27 \, x^{4} + x^{3} + 9 \, x^{2} + 27 \, x + 27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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mupad [B]  time = 0.05, size = 27, normalized size = 1.00 \[ \frac {-5\,x^6+x^4+5\,x^2-3\,x+2}{{\left (x^4+x+3\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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

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sympy [B]  time = 0.32, size = 60, normalized size = 2.22 \[ \frac {- 5 x^{6} + x^{4} + 5 x^{2} - 3 x + 2}{x^{12} + 3 x^{9} + 9 x^{8} + 3 x^{6} + 18 x^{5} + 27 x^{4} + x^{3} + 9 x^{2} + 27 x + 27} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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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